Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ1 due 01/01/2007 at 02:00am EST.
4.
1. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 1.pg
Match each of the following differential equations with a solution from the list below.
1. 2x2 y00 + 3xy0 = y
2. y00 + 6y0 + 8y = 0
3. y00 + y = 0
4. y00 − 6y0 + 8y = 0
1
A. y =
x
B. y = e4x
C. y = cos(x)
D. y = e−2x
5.
A.
B.
C.
D.
E.
dy
= 10xy
dx
dy
+ 21x2 y = 21x2
dx
y = Ae−7x + Bxe−7x
y = A cos(5x) + B sin(5x)
3yx2 − 5y3 = C
2
y = Ae5x
3
y = Ce−7x + 1
5. (1 pt) rochesterLibrary/setDiffEQ1/ur de 1 2.pg
Just as there are simultaneous algebraic equations (where a pair
of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to
satisfy a pair of differential equations).
Indicate which pairs of functions satisfy this system. It will take
some time to make all of the calculations.
y01 = y1 − 2y2
y02 = 3y1 − 4y2
2. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 1a.pg
Match each of the differential equation with its solution.
1. xy0 − y = x2
2. y00 + y = 0
3. 2x2 y00 + 3xy0 = y
4. y00 + 9y0 + 18y = 0
A. y = sin(x)
B. y = e−6x
1
C. y = x 2
D. y = 3x + x2
•
•
•
•
•
•
•
3. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 2.pg
Match each differential equation to a function which is a solution.
FUNCTIONS
A. y = 3x + x2 ,
B. y = e−5x ,
C. y = sin(x),
1
D. y = x 2 ,
E. y = 3 exp(8x),
DIFFERENTIAL EQUATIONS
1. 2x2 y00 + 3xy0 = y
2. y0 = 8y
3. y00 + y = 0
4. xy0 − y = x2
A. y1 = ex
y2 = ex
−2x
B. y1 = 2e
y2 = 3e−2x
−x
C. y1 = e
y2 = e−x
D. y1 = cos(x)
y2 = − sin(x)
E. y1 = sin(x)
y2 = cos(x)
F. y1 = sin(x) + cos(x)
y2 = cos(x) − sin(x)
G. y1 = e4x
y2 = e4x
As you can see, finding all of the solutions, particularly of a
system of equations, can be complicated and time consuming.
It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions we will
be able to predict the rest of the solutions using the structure of
the family of solutions.
6. (1 pt) rochesterLibrary/setDiffEQ1/ur de 1 1.pg
It can be helpful to classify a differential equation, so that we
can predict the techniques that might help us to find a function
which solves the equation. Two classifications are the order
of the equation – (what is the highest number of derivatives
involved) and whether or not the equation is linear .
Linearity is important because the structure of the the family of
solutions to a linear equation is fairly simple. Linear equations
can usually be solved completely and explicitly.
Determine whether or not each equation is linear:
d2y
dy
? 1. t 2 2 + t + 2y = sint
dt
dt
d 4 y d 3 y d 2 y dy
? 2.
+
+
+
=1
dt 4 dt 3 dt 2 dt
3
dy
d y
? 3.
+ t + (cos2 (t))y = t 3
dt 3
dt
dy
2
? 4.
+ ty = 0
dt
4. (1 pt) rochesterLibrary/setDiffEQ1/osu de 1 3.pg
Match the following differential equations with their solutions.
The symbols A, B, C in the solutions stand for arbitrary constants.
You must get all of the answers correct to receive credit.
d2y
+ 25y = 0
1.
dx2
dy
−2xy
2.
= 2
dx x − 5y2
d2y
dy
3.
+ 14 + 49y = 0
dx2
dx
1
13. (1 pt) rochesterLibrary/setDiffEQ1/ur de 1 4.pg
Find the value of k for which the constant function x(t) = k is a
dx
solution of the differential equation 7t 5 + 9x − 5 = 0.
dt
7. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 4.pg
It is easy to check that for any value of c, the function
c
y = x2 + 2
x
is solution of equation
0
14. (1 pt) rochesterLibrary/setDiffEQ1/ur de 1 5.pg
Which of the following functions are solutions of the differential
equation y00 − 14y0 + 49y = 0?
• A. y(x) = x2 e7x
• B. y(x) = xe7x
• C. y(x) = 7xe7x
• D. y(x) = 0
• E. y(x) = e−7x
• F. y(x) = e7x
• G. y(x) = 7x
2
xy + 2y = 4x , (x > 0).
Find the value of c for which the solution satisfies the initial
condition y(9) = 8.
c=
8. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 4a.pg
The functions
c
y = x2 + 2
x
are all solutions of equation:
xy0 + 2y = 4x2 , (x > 0).
15. (1 pt) rochesterLibrary/setDiffEQ1/dp7 1 1.pg
Consider the curves in the first quadrant that have equations
Find the constant c which produces a solution which also satisfies the initial condition y(10) = 7.
c=
9. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 3.pg
It is easy to check that for any value of c, the function
y = A exp(6x),
where A is a positive constant.
Different values of A give different curves. The curves form a
family, F.
Let P = (6, 1). Let C be the member of the family F that goes
through P.
A. Let y = f (x) be the equation of C. Find f (x).
f (x) =
B. Find the slope at P of the tangent to C.
slope =
C. A curve D is perpendicular to C at P. What is the slope of the
tangent to D at the point P? slope =
D. Give a formula g(y) for the slope at (x, y) of the member of
F that goes through (x, y). The formula should not involve A or
x.
g(y) =
E. A curve which at each of its points is perpendicular to the
member of the family F that goes through that point is called
an orthogonal trajectory to F. Each orthogonal trajectory to F
satisfies the differential equation
dy
1
=−
,
dx
g(y)
where g(y) is the answer to part D.
Find a function h(y) such that x = h(y) is the equation of the
orthogonal trajectory to F that passes through the point P.
h(y) =
y = ce−2x + e−x
is solution of equation
y0 + 2y = e−x .
Find the value of c for which the solution satisfies the initial
condition y(−5) = 9.
c=
10. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 3a.pg
The family of functions
y = ce−2x + e−x
is solution of the equation
y0 + 2y = e−x .
Find the constant c which defines the solution which also satisfies the initial condition y(2) = 10.
c=
11. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 5.pg
Find the two values of k for which
y(x) = ekx
is a solution of the differential equation
y00 − 1y0 + 0y = 0.
smaller value =
larger value =
16. (1 pt) rochesterLibrary/setDiffEQ1/dp7 1 2.pg
The solution of a certain differential equation is of the form
y(t) = a exp(5t) + b exp(9t),
12. (1 pt) rochesterLibrary/setDiffEQ1/ur de 1 3.pg
Find all values of k for which the function y = sin(kt) satisfies
the differential equation y00 + 14y = 0. Separate your answers by
commas.
where a and b are constants.
The solution has initial conditions y(0) = 1 and y0 (0) = 1.
Find the solution by using the initial conditions to get linear
equations for a and b.
y(t) =
2
c
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Team, Department of Mathematics, University of Rochester
3
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ10Linear2ndOrderNonhom due 01/10/2007 at 02:00am EST.
1.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 4.pg
7.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 10.pg
Find a single solution of y if y00 = 2.
y=
Find a particular solution to
2.
y00 + 8y0 + 16y = 3.5e−4t .
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom-
yp =
/ur de 10 5.pg
Use the method of undetermined coefficients to find one solution of
y00 − 10y0 + 28y = 6e7t .
y=
(It doesn’t matter which specific solution you find for this problem.)
8.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 8.pg
3.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 13.pg
9.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 12.pg
Use the method of undetermined coefficients to find one solution of
y00 + 4 y0 − 4 y = (6t 2 + 9t − 1) e3t .
Note that the method finds a specific solution, not the general
one.
y=
Find a particular solution to
Find a particular solution to the differential equation
−4y00 + 0y0 + 1y = −2t 2 + 1t + 4e4t .
yp =
y00 + 5y0 + 4y = 12te3t .
yp =
10. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 9.pg
Find a particular solution to
4.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 6.pg
y00 + 25y = −50 sin(5t).
Use the method of undetermined coefficients to find one solution of
y00 − 12y0 + 86y = 48e6t cos(7t) + 48e6t sin(7t) + 1e3t .
(It doesn’t matter which specific solution you find for this problem.)
y=
yp =
11. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 1.pg
Find the solution of
y00 + 4y0 + 3y = 6 e0t
with y(0) = 2 and y0 (0) = 9.
y=
5.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 7.pg
Use the method of undetermined coefficients to find one solution of
y00 + 2y0 + 2y = (10t + 7) e−t cos(t) + (11t + 25) e−t sin(t).
(It doesn’t matter which specific solution you find for this problem.)
y=
6.
12. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 2.pg
Find the solution of
y00 + 4y0 + 4y = 54 e1t
with y(0) = 5 and y0 (0) = 6.
y=
13. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 3.pg
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom-
Find the solution of
/ur de 10 11.pg
Find a particular solution to the differential equation
00
0
y00 + 4y0 = 256 sin(4t) + 128 cos(4t)
3
y − 5y + 4y = −48t .
with y(0) = 9 and y0 (0) = 5.
y=
yp =
1
14. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 14.pg
16.
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom-
/ur de 10 16.pg
Find y as a function of x if
Find a particular solution to
x2 y00 − 3xy0 − 32y = x5 ,
y(1) = 7, y0 (1) = −7.
y=
15.
y00 − 4y0 + 4y =
6.5e2t
t2 + 1
.
yp =
(1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom-
/ur de 10 15.pg
17. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom/ur de 10 17.pg
Find y as a function of x if
x2 y00 − 5xy0 + 9y = x2 ,
Find a particular solution to y00 + 16y = −8 sec(4t).
yp =
y(1) = 8, y0 (1) = −4.
y=
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ11ModelingWith2ndOrder due 01/11/2007 at 02:00am EST.
Calculate the specific solution that has initial conditions t = 0
and w(0) = 2.5.
w(t) =
1.
(1 pt) rochesterLibrary/setDiffEQ11ModelingWith2ndOrder/ur de 11 1.pg
Another ”realistic” problem:
The following problem is similar to the problem in an earlier
assignment about the bank account growing with periodic deposits. The basic procedure for this problem is not too hard, but
getting details of the calculation correct is NOT easy, and may
take some time.
A ping-pong ball is caught in a vertical plexiglass column in
which the air flow alternates sinusoidally with a period of 60
seconds. The air flow starts with a maximum upward flow at the
rate of 1.2m/s and at t = 30 seconds the flow has a minimum
(upward) flow of rate of −3.2m/s. (To make this clear: a flow
of −5m/s upward is the same as a flow downward of 5m/s.
The ping-pong ball is subjected to the forces of gravity
(−mg) where g = 9.8m/s2 and forces due to air resistance which
are equal to k times the apparent velocity of the ball through the
air.
What is the average velocity of the air flow? You can average the velocity over one period or over a very long time – the
answer should come out about the same – right?
. (Include units.)
Write a formula for the velocity of the air flow as a function
of time.
A(t) =
Write the differential equation satisfied by the velocity of
the ping-pong ball (relative to the fixed frame of the plexiglass
tube.) The formulas should not have units entered, but use units
to trouble shoot your answers. Your answer can include the parameters m - the mass of the ball and k the coefficient of air
resistance, as well as time t and the velocity of the ball v. (Use
just v, not v(t) the latter confuses the computer.)
v0 (t) =
Think about what effect increasing the mass has on the amplitude, on the phase shift? Does this correspond with your expectations?
2.
(1 pt) rochesterLibrary/setDiffEQ11ModelingWith2ndOrder-
/ur de 11 2.pg
A steel ball weighing 128 pounds is suspended from a spring.
This stretches the spring 128
17 feet.
The ball is started in motion from the equilibrium position
with a downward velocity of 7 feet per second.
The air resistance (in pounds) of the moving ball numerically
equals 4 times its velocity (in feet per second) .
Suppose that after t seconds the ball is y feet below its rest
position. Find y in terms of t. (Note that this means that the
postiive direction for y is down.)
y=
Take as the gravitational acceleration 32 feet per second per
second.
3.
(1 pt) rochesterLibrary/setDiffEQ11ModelingWith2ndOrder/ur de 11 3.pg
A hollow steel ball weighing 4 pounds is suspended from a
spring. This stretches the spring 21 feet.
The ball is started in motion from the equilibrium position
with a downward velocity of 5 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times
its velocity (in feet per second) .
Suppose that after t seconds the ball is y feet below its rest
position. Find y in terms of t. (Note that the positive direction
is down.)
Take as the gravitational acceleration 32 feet per second per
second.
y=
Use the method of undetermined coefficients to find one periodic solution to this equation:
v(t) =
4.
(1 pt) rochesterLibrary/setDiffEQ11ModelingWith2ndOrder/ur de 11 4.pg
Find the amplitude and phase shift of this solution. You do
not need to enter units.
v(t) =
cos(
∗t −
Find the general solution, by adding on a solution to the homogeneous equation. Notice that all of these solutions tend towards the periodically oscillating solution. This is a generalization of the notion of stability that we found in autonomous
differential equations.
This problem is an example of critically damped harmonic mo) tion.
A hollow steel ball weighing 4 pounds is suspended from a
spring. This stretches the spring 81 feet.
The ball is started in motion from the equilibrium position
with a downward velocity of 7 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times
1
y=
its velocity (in feet per second) . Suppose that after t seconds
the ball is y feet below its rest position. Find y in terms of t.
Take as the gravitational acceleration 32 feet per second per
second. (Note that the positive y direction is down in this problem.)
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ12HigherOrder due 01/12/2007 at 02:00am EST.
y(x) =
1. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 1.pg
Match the third order linear equations with their fundamental
solution sets.
1. ty000 − y00 = 0
2. y000 + 3y00 + 3y0 + y = 0
3. y000 + y0 = 0
4. y000 − 8y00 + y0 − 8y = 0
5. y000 − 7y00 + 12y0 = 0
6. y000 − y00 − y0 + y = 0
A.
B.
C.
D.
E.
F.
4. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 4.pg
Find y as a function of x if
y(4) − 6y000 + 9y00 = 0,
y(0) = 13, y0 (0) = 4, y00 (0) = 9, y000 (0) = 0.
y(x) =
5. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 5.pg
Find y as a function of x if
y000 − 5y00 − y0 + 5y = 0,
y(0) = −9, y0 (0) = 7, y00 (0) = 87.
y(x) =
1, t, t 3
e8t , cos(t), sin(t)
et , tet , e−t
e−t , te−t , t 2 e−t
1, cos(t), sin(t)
1, e4t , e3t
6. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 6.pg
If L = D2 + 3xD − 2x and y(x) = 2x − 5e4x , then
Ly =
7. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 7.pg
Find y as a function of x if
y000 − 8y00 + 15y0 = 8ex ,
0
y(0) = 14, y (0) = 28, y00 (0) = 29.
y(x) =
2. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 2.pg
Find y as a function of x if
y000 − 11y00 + 24y0 = 0,
0
y(0) = 3, y (0) = 3, y00 (0) = 6.
y(x) =
8. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 8.pg
Find y as a function of x if
y(4) − 10y000 + 25y00 = −196e−2x ,
0
y(0) = 12, y (0) = 13, y00 (0) = 21, y000 (0) = 8.
y(x) =
3. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 3.pg
Find y as a function of x if
y000 + 64y0 = 0,
0
y(0) = −3, y (0) = 16, y00 (0) = −128.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ13Systems1stOrder due 01/13/2007 at 02:00am EST.
4.
1. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 1.pg
Write the given second order equation as its equivalent system
of first order equations.
(1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 4.pg
u00 + 3u0 + 3u = 0
Use v to represent the ”velocity function”, i.e. v = u0 (t).
Use v and u for the two functions, rather than u(t) and v(t). (The
latter confuses webwork. Functions like sin(t) are ok.)
u0 =
v0 =
Now
write
matrices:
the
system using
d u
u
=
.
v
dt v
Consider two interconnected tanks as shown in the figure
above. Tank 1 initial contains 100 L (liters) of water and 435
g of salt, while tank 2 initially contains 60 L of water and 395 g
of salt. Water containing 45 g/L of salt is poured into tank1 at a
rate of 2 L/min while the mixture flowing into tank 2 contains a
salt concentration of 25 g/L of salt and is flowing at the rate of 4
L/min. The two connecting tubes have a flow rate of 5.5 L/min
from tank 1 to tank 2; and of 3.5 L/min from tank 2 back to tank
1. Tank 2 is drained at the rate of 6 L/min.
You may assume that the solutions in each tank are thoroughly mixed so that the concentration of the mixture leaving
any tank along any of the tubes has the same concentration of
salt as the tank as a whole. (This is not completely realistic, but
as in real physics, we are going to work with the approximate,
rather than exact description. The ’real’ equations of physics are
often too complicated to even write down precisely, much less
solve.)
How does the water in each tank change over time?
Let p(t) and q(t) be the amount of salt in g at time t in tanks
1 and 2 respectively. Write differential equations for p and q.
(As usual, use the symbols p and q rather than p(t) and q(t).)
p0 =
2. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 2.pg
Write the given second order equation as its equivalent system
of first order equations.
u00 + 2.5u0 + 2u = 6 sin(3t),
u(1) = −5.5,
u0 (1) = −4
Use v to represent the ”velocity function”, i.e. v = u0 (t).
Use v and u for the two functions, rather than u(t) and v(t). (The
latter confuses webwork. Functions like sin(t) are ok.)
u0 =
v0 =
Now
write
the
system using matrices:
d u
u
=
+
v
dt v
and the initial
vector valued function is:
value for the
u(1)
.
=
v(1)
q0 =
3. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 3.pg
Write the given second order equation as its equivalent system
of first order equations.
Give the initial values:
p(0)
=
.
q(0)
Show the equation that needs to be solved to find a constant
solution to the
equation: differential
p
=
.
q
A constant solution is obtained if p(t) =
for all time t
and q(t) =
for all time t.
t 2 u00 + 2.5tu0 + (t 2 + 7)u = 4.5 sin(3t)
Use v to represent the ”velocity function”, i.e. v = u0 (t).
Use v and u for the two functions, rather than u(t) and v(t). (The
latter confuses webwork. Functions like sin(t) are ok.)
u0 =
v0 =
Now
write
the
system using matrices:
d u
u
=
+
.
v
dt v
5. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 5.pg
Match the differential equations and their matrix function solutions:
1
It’s good practice to multiply at least one matrix solution out
fully, to make sure that you know how to do it, but you can
get the other answers quickly by process of elimination and just
multiply out one row or one column.


-86 218 -160
-49
80  y(t)
1. y0 (t) = 73
 111 -138 165

15 0
0
2. y0 (t) = 4 20 -15  y(t)
 4 30 -25
-2 0 0
3. y0 (t) = 0 -2 0  y(t)
0 0 -2

A. y(t) =

B. y(t) =

C. y(t) =
7. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 7.pg
Calculate the eigenvalues of this matrix:
[Note– you’ll probably want to use a graphing calculator to
estimate the roots of the polynomial which defines the eigenvalues. You can use the web version at xFunctions.
If you select the ”integral curves utility” from the main menu,
will also be able to plot the integral curves of the associated diffential equations.
]
-20 0
A=
0 0
smaller eigenvalue =
associated eigenvector = (
,
)
larger eigenvalue =
associated, eigenvector = (
,
)

−5e−2t 2e−2t 3e−2t
−2e−2t 2e−2t 2e−2t 
e−2t
0
4e−2t

15t
5e
0
0
2e15t e5t 2e10t 
2e15t e5t 4e10t

e60t
−2e45t 4e−75t
−3e60t
e45t
−2e−75t 
60t
45t
−5e
3e
−3e−75t
If y0 = Ay is a differential equation, how would the solution
curves behave?
• A. All of the solutions curves would converge towards
0. (Stable node)
• B. The solution curves converge to different points.
• C. The solution curves would race towards zero and
then veer away towards infinity. (Saddle)
• D. All of the solution curves would run away from 0.
(Unstable node)
6. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 6.pg
Match the differential equations and their vector valued function
solutions:
It will be good practice to multiply at least one solution out fully,
to make sure that you know how to do it, but you can get the
other answers quickly by process of elimination and just multiply out one row element.


14 0 -4
1. y0 (t) = 2 13 -8  y(t)
 -3 0 25 
-97 33 -5
2. y0 (t) = -140 84 35  y(t)
-4
15 -8


-86 218 -160
-49
80  y(t)
3. y0 (t) = 73
111 -138 165

8. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 8.pg
Calculate the eigenvalues of this matrix:
[Note– you’ll probably want to use a graphing calculator to
estimate the roots of the polynomial which defines the eigenvalues. You can use the web version at xFunctions.
If you select the ”integral curves utility” from the main menu,
will also be able to plot the integral curves of the associated diffential equations.
] -60 -70
A=
56 108
smaller eigenvalue =
associated eigenvector = (
,
)
larger eigenvalue =
associated, eigenvector = (
,
)
If y0 = Ay is a differential equation, how would the solution
curves behave?

-2
A. y(t) = -4  e−21t
 4 
-2
B. y(t) = 1  e45t
3
 
4
C. y(t) = 5  e13t
1
• A. All of the solutions curves would converge towards
0. (Stable node)
• B. All of the solution curves would run away from 0.
(Unstable node)
• C. The solution curves would race towards zero and
then veer away towards infinity. (Saddle)
• D. The solution curves converge to different points.
2
13.
9. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/ur de 13 9.pg
Consider the following model for the populations of rabbits and
wolves (where R is the population of rabbits and W is the population of wolves).
dR
dt
= 0.16R(1 − 0.0004R) − 0.002RW
dW
dt
= −0.04W + 8e − 05RW
(1
pt)
rochesterLibrary/setDiffEQ13Systems1stOrder-
Solvethe system dx
-12 5
=
x
-30 13
dt
14.
8
with the initial value x(0) =
.
21
x(t) =
.
(1
pt)
rochesterLibrary/setDiffEQ13Systems1stOrder-
15.
Consider the interaction of two species of animals in a habitat.
We are told that the change of the populations x(t) and y(t) can
be modeled by the equations
dy
dt
= 2.5x − 1.2y.
(1
pt)
rochesterLibrary/setDiffEQ13Systems1stOrder-
(1
pt)
rochesterLibrary/setDiffEQ13Systems1stOrder-
Solve the system
dx
-3 -3
=
x
6 3 dt
2
with x(0) =
.
3
Give your solution in real form.
x1 =
,
x2 =
.
rochesterLibrary/setDiffEQ13Systems1stOrder-
= 1.6x − 0.8y,
= m21 b + m22 s.
/ur de 13 15.pg
/ur de 13 12.pg
dx
dt
ds
dt
? 1. Describe the trajectory.
13
with the initial value x(0) =
.
15
x(t) =
.
pt)
= m11 b + m12 s,
Solve the system
dx
-2 -2
=
x
2 -2 dt
1
with x(0) =
.
1
Give your solution in real form.
x1 =
,
x2 =
.
Solvethe system dx
-12 -12
=
x
-9
-9
dt
(1
db
dt
/ur de 13 14.pg
/ur de 13 11.pg
12.
rochesterLibrary/setDiffEQ13Systems1stOrder-
,
m11 =
m12 =
,
,
m21 =
m22 =
.
(b) Find b(t) and s(t).
b(t) =
,
s(t) =
.
/ur de 13 10.pg
11.
pt)
David opens a bank account with an initial balance of 500 dollars. Let b(t) be the balance in the account at time t. Thus
b(0) = 500. The bank is paying interest at a continuous rate of
4% per year. David makes deposits into the account at a continuous rate of s(t) dollars per year. Suppose that s(0) = 500 and
that s(t) is increasing at a continuous rate of 6% per year (David
can save more as his income goes up over time).
(a) Set up a linear system of the form
Find all the equilibrium solutions:
(a) In the absence of wolves, the population of rabbits ap.
proaches
(b) In the absence of rabbits, the population of wolves approaches
.
(c) If both wolves and rabbits are present, their populations apand w =
.
proach r =
10.
(1
/ur de 13 13.pg
? 1. Describe the trajectory.
16.
(1
pt)
rochesterLibrary/setDiffEQ13Systems1stOrder-
/ur de 13 16.pg
Multiplying the differential equation
df
+ a f (t) = g(t),
dt
where a is a constant and g(t) is a smooth function, by eat , gives
? 1. What kind of interaction do we observe?
3
eat
df
+ eat a f (t) = eat g(t),
dt
d at
e f (t) = eat g(t),
dt
at
Z
e f (t) =
f (t) = e−at
Z
Use this
to solve
the initial value problem
dx
3 1
=
x,
0 2
dt
-5
with x(0) =
,
4
i.e. find first x2 (t) and then x1 (t).
,
x1 (t) =
x2 (t) =
.
eat g(t)dt,
eat g(t)dt.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
4
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ2DirectionFields due 01/02/2007 at 02:00am EST.
1.
(1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 1/ur de 2 1.pg
Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you
can probably solve this problem by guessing, it is useful to try
to predict characteristics of the direction field and then match
them to the picture. Here are some handy characteristics to start
with – you will develop more as you practice.
A. Set y equal to zero and look at how the derivative behaves along the x-axis.
B. Do the same for the y-axis by setting x equal to 0
C. Consider the curve in the plane defined by setting y0 = 0
– this should correspond to the points in the picture
where the slope is zero.
D. Setting y0 equal to a constant other than zero gives the
curve of points where the slope is that constant. These
are called isoclines, and can be used to construct the
direction field picture by hand.
Go to this page to launch the phase plane plotter to check your
answers. (Choose the ”integral curves utility” from the applet
menu, enter dx/dt = 1 to identify the variables x and t and then
enter the function you want for dy/dx = dy/dt = . . . ).
1. y0 = −2 + x − y
2. y0 = e−x − y
3. y0 = 3 sin(x) + 1 + y
A
B
A
B
C
D
3. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 3.pg
Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you
can probably solve this problem by guessing, it is useful to try
to predict characteristics of the direction field and then match
them to the picture.
Here are some handy characteristics to start with – you will develop more as you practice.
A. Set y equal to zero and look at how the derivative behaves along the x-axis.
B. Do the same for the y-axis by setting x equal to 0
C. Consider the curve in the plane defined by setting y0 = 0
– this should correspond to the points in the picture
where the slope is zero.
D. Setting y0 equal to a constant other than zero gives the
curve of points where the slope is that constant. These
are called isoclines, and can be used to construct the
direction field picture by hand.
C
2.
(1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 2/ur de 2 2.pg
This problem is harder, and doesn’t give you clues as to which
matches you have right. Study the previous problem, if you are
having trouble.
Go to this page to launch the phase plane plotter to check your
answers.
Match the following equations with their direction field. Clicking on each picture will give you an enlarged view.
1. y0 = 2x − 1 − y2
2. y0 = y(4 − y)
3. y0 = x + 2y
y3
x3
4. y0 = − y −
6
6
2
1. y0 = 2xy + 2xe−x
2. y0 = 2y − 2
(2x + y)
3. y0 = −
(2y)
4. y0 = 2 sin(3x) + 1 + y
1
A
B
C
D
5. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 5.pg
Use Euler’s method with step size 0.5 to compute the approximate y-values y1 , y2 , y3 , and y4 of the solution of the initial-value
problem
y0 = −2 − 2x − 4y, y(0) = 2.
C
y1 =
y2 =
y3 =
y4 =
D
6. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 6.pg
Use Euler’s method with step size 0.3 to estimate y(1.5), where
y(x) is the solution of the initial-value problem
4. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 4.pg
Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you
can probably solve this problem by guessing, it is useful to try
to predict characteristics of the direction field and then match
them to the picture. Here are some handy characteristics to start
with – you will develop more as you practice.
A. Set y equal to zero and look at how the derivative behaves along the x-axis.
B. Do the same for the y-axis by setting x equal to 0
C. Consider the curve in the plane defined by setting y0 = 0
– this should correspond to the points in the picture
where the slope is zero.
D. Setting y0 equal to a constant other than zero gives the
curve of points where the slope is that constant. These
are called isoclines, and can be used to construct the
direction field picture by hand.
y0 = −5x + y2 , y(0) = 0.
y(1.5) =
.
7. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 7.pg
Suppose you have just poured a cup of freshly brewed coffee
with temperature 90◦C in a room where the temperature is 20◦C.
Newton’s Law of Cooling states that the rate of cooling of an
object is proportional to the temperature difference between the
object and its surroundings. Therefore, the temperature of the
coffee, T (t), satisfies the differential equation
dT
= k(T − Troom )
dt
where Troom = 20 is the room temperature, and k is some constant.
Suppose it is known that the coffee cools at a rate of 1◦C per
minute when its temperature is 60◦C.
A. What is the limiting value of the temperature of the coffee?
lim T (t) =
t→∞
B. What is the limiting value of the rate of cooling?
dT
lim
=
t→∞ dt
C. Find the constant k in the differential equation.
k=
.
D. Use Euler’s method with step size h = 3 minutes to estimate
the temperature of the coffee after 15 minutes.
T (10) =
.
1. y0 = 2y + x2 e2x
(2x + y)
2. y0 = −
(2y)
3. y0 = y + 2
4. y0 = 2 sin(x) + 1 + y
2
A
,
,
,
.
B
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
3
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ3Separable due 01/03/2007 at 02:00am EST.
6. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 16.pg
Solve the separable differential equation
dx
1
= x2 + ,
dt
9
and find the particular solution satisfying the initial condition
1. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 11.pg
Solve the separable differential equation
dy
= −6y,
dx
and find the particular solution satisfying the initial condition
x(0) = −9.
y(0) = 4.
y(x) =
x(t) =
.
7. (1 pt) rochesterLibrary/setDiffEQ3Separable/osu de 3 11.pg
Find the particular solution of the differential equation
dy
= (x − 8)e−2y
dx
satisfying the initial condition y(8) = ln(8).
Answer: y =
.
Your answer should be a function of x.
8. (1 pt) rochesterLibrary/setDiffEQ3Separable/osu de 3 12.pg
Find the particular solution of the differential equation
2. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 12.pg
Solve the separable differential equation
dx 4
= ,
dt
x
and find the particular solution satisfying the initial condition
x(0) = 1.
x(t) =
.
x2 dy
1
=
2
y − 7 dx 2y
√
satisfying the initial condition y(1) = 8.
Answer: y =
.
Your answer should be a function of x.
9. (1 pt) rochesterLibrary/setDiffEQ3Separable/jas7 4 5.pg
Find u from the differential equation and initial condition.
du
= e3.5t−1.6u , u(0) = 1.
dt
u=
.
10. (1 pt) rochesterLibrary/setDiffEQ3Separable/jas7 4 5a.pg
Solve the separable differential equation for u
du
= e4u+7t .
dt
Use the following initial condition: u(0) = 9.
u=
.
11. (1 pt) rochesterLibrary/setDiffEQ3Separable/jas7 4 5b.pg
Solve the separable differential equation for u
du
= e3u+9t .
dt
Use the following initial condition: u(0) = −8.
.
u=
12. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 10.pg
Solve the separable differential equation
p
dy
11x − 4y x2 + 1
= 0.
dx
Subject to the initial condition: y(0) = 7.
y=
.
3. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 13.pg
Solve the separable differential equation
dy
= −8y6 ,
dt
and find the particular solution satisfying the initial condition
y(0) = −5.
y(t) =
.
4. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 14.pg
Solve the separable differential equation
p
y0 (x) = 2y(x) + 37,
and find the particular solution satisfying the initial condition
y(−3) = 6.
y(x) =
.
5. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 15.pg
Solve the separable differential equation
dy
−0.8
=
,
dx cos(y)
and find the particular solution satisfying the initial condition
y(0) =
y(x) =
.
π
.
4
.
1
13. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 13.pg
Find f (x) if y = f (x) satisfies
21. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 8b.pg
Find the function y = y(x) (for x > 0 ) which satisfies the separable differential equation
dy
= 30yx2
dx
and the y-intercept of the curve y = f (x) is 6.
.
f (x) =
dy 7 + 20x
; x>0
=
dx
xy2
14. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 13a.pg
Find an equation of the curve that satisfies
with the initial condition y(1) = 6.
.
y=
dy
= 54yx5
dx
and whose y-intercept is 5.
y(x) =
.
22. (1 pt) rochesterLibrary/setDiffEQ3Separable/osu de 3 13.pg
Find the solution of the differential equation
15. (1 pt) rochesterLibrary/setDiffEQ3Separable/osu de 3 14.pg
Find the solution of the differential equation
dy
x
3e7x
= −49 2
dx
y
which satisfies the initial condition y(0) = 1.
y=
.
which satisfies the initial condition y(1) = e2 .
y=
.
(ln(y))7
23. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 1.pg
A. Solve the following initial value problem:
16. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 3.pg
Find a function y of x such that
(t 2 − 18t + 72)
0
6yy = x and y(6) = 11.
y=
dy
= x7 y
dx
.
dy
=y
dt
with y(9) = 1. (Find y as a function of t.)
y=
.
B. On what interval is the solution valid?
Answer: It is valid for
<t <
.
C. Find the limit of the solution as t approaches the left end of
the interval.
(Your answer should be a number or the word ”infinite”.)
Answer:
.
D. Similar to C, but for the right end.
Answer:
.
17. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 17.pg
Find k such that x(t) = 5t is a solution of the differential equadx
tion
= kx.
dt
k=
.
18. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 3a.pg
Solve the seperable differential equation
10yy0 = x.
Use the following initial condition: y(10) = 3.
Express x2 in terms of y.
(function of y).
x2 =
24. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 2.pg
The differential equation
19. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 8.pg
Solve the differential equation
dy
y2 + 10y + 25
= cos(x)
dx
5y + 25
dy
= 1 + x.
dx
Use the initial condition y(1) = 5.
Express y13 in terms of x.
y13 =
.
(y12 x)
has an implicit general solution of the form F(x, y) = K.
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
F(x, y) = G(x) + H(y) = K.
20. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns7 4 8a.pg
Solve the seperable differential equation for.
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
dy 1 + x
=
; x>0
dx
xy7
Use the following initial condition: y(1) = 5.
y8 =
.
.
2
<x<
25. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 3.pg
The differential equation
dy
36
= (16 − x2 )−1/2 exp(−6y)
dx
has an implicit general solution of the form F(x, y) = K.
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
.
29. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 7.pg
The differential equation
6x + 6
dy
=
dx 15y2 + 6y + 6
has an implicit general solution of the form F(x, y) = K.
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
F(x, y) = G(x) + H(y) = K.
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
F(x, y) = G(x) + H(y) = K.
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
.
26. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 4.pg
The differential equation
dy
14
= 1/8
dx y + 4 x2 y1/8
.
30. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 8.pg
The differential equation
has an implicit general solution of the form F(x, y) = K.
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
exp(y)
F(x, y) = G(x) + H(y) = K.
dy
14x + 2
=
dx 2 sin(y) + 8 cos(y)
has an implicit general solution of the form F(x, y) = K.
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
F(x, y) = G(x) + H(y) = K.
.
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
27. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 5.pg
The differential equation
dy
(15 + 8 cos(x)) = sin(x) cos(y)
dx
has an implicit general solution of the form F(x, y) = K.
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
.
31. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 9.pg
A. Solve the following initial value problem:
F(x, y) = G(x) + H(y) = K.
cos(t)2
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
dy
=1
dt
with y(21) = tan(21).
(Find y as a function of t.)
.
y=
B. On what interval is the solution valid?
(Your answer should involve pi.)
Answer: It is valid for
<t <
.
C. Find the limit of the solution as t approaches the left end of
the interval. (Your answer should be a number or ”PINF” or
”MINF”.
”PINF” stands for plus infinity and ”MINF” stands for minus
infinity.)
.
Answer:
D. Similar to C, but for the right end.
Answer:
.
.
28. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 6.pg
A. Find y in terms of x if
dy
= x2 y−3
dx
and y(0) = 8.
y(x) =
.
B. For what x-interval is the solution defined?
(Your answers should be numbers or plus or minus infinity. For
plus infinity enter ”PINF”; for minus infinity enter ”MINF”.)
The solution is defined on the interval:
3
In fact, because the differential equation is separable, we can
define the solution curve implicitly by a function in the form
32. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 10.pg
The differential equation
F(x, y) = G(x) + H(y) = K.
Find such a solution and then give the related functions requested.
F(x, y) = G(x) + H(y) =
dy
= 8 + 6 x + 16 y + 12 xy
dx
has an implicit general solution of the form F(x, y) = K.
.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
4
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ4Linear1stOrder due 01/04/2007 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/osu de 4 14.pg
Find the particular solution of the differential equation
8. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 7.pg
Solve the initial value problem
dy
10(t + 1) − 9y = 9t,
dt
for t > −1 with y(0) = 3.
.
y=
dy
+ 3y = 7
dx
satisfying the initial condition y(0) = 0.
.
Answer: y =
Your answer should be a function of x.
9. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 15.pg
Solve the initial value problem
2. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 16.pg
Find the function satisfying the differential equation
dx
− 3x = cos(5t)
dt
f 0 (t) − f (t) = −2t
and the condition f (3) = 3.
.
f (t) =
with x(0) = 5.
x(t) =
3. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 1.pg
GUESS one function y(t) which solves the problem below, by
determining the general form the function might take and then
evaluating some coefficients.
dy
7t + y = t 3
dt
Find y(t).
y(t) =
.
10. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/osu de 4 15.pg
Find the particular solution of the differential equation
dy
+ y cos(x) = 5 cos(x)
dx
satisfying the initial condition y(0) = 7.
Answer: y(x)=
.
11. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 4.pg
Solve the initial value problem
dy
− y = 3 exp(t) + 21 exp(4t)
dt
with y(0) = 7.
y=
.
4. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 2.pg
GUESS one function y(t) which solves the problem below, by
determining the general form the function might take and then
evaluating some coefficients.
dy
+ 7y = exp(2t)
dt
Find y(t).
y(t) =
.
12. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 8.pg
Solve the initial value problem
dy
+ 2y = 40 sin(t) + 25 cos(t)
dt
with y(0) = 7.
y=
.
5. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 3.pg
Find the function satisfying the differential equation
y0 − 5y = 6e8t
and y(0) = −1.
y=
.
13. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 11.pg
Solve the following initial value problem:
dy
9 + y = 63t
dt
with y(0) = 7.
(Find y as a function of t.)
y=
.
.
6. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 5.pg
Solve the following initial value problem:
dy
t + 9y = 4t
dt
with y(1) = 1.
y=
.
14. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 13.pg
Solve the initial value problem
dy
8(sin(t) + (cost)y) = (cos(t))(sin(t))6 ,
dt
for 0 < t < π and y(π/2) = 16.
y=
.
7. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 6.pg
Solve the following initial value problem:
dy
+ 0.7ty = 8t
dt
with y(0) = 8.
y=
.
1
15. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 14.pg
Find the function y(t) that satisfies the differential equation
19. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 17.pg
A function y(t) satisfies the differential equation
2
dy
− 2ty = −6t 2 et
dt
and the condition y(0) = 5.
y(t) =
.
dy
= −y4 − 2y3 + 15y2 .
dt
(a) What are the constant solutions of this equation?
Separate your answers by commas.
.
(b) For what values of y is y increasing?
<y<
.
16. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 9.pg
A. Let g(t) be the solution of the initial value problem
dy
5t + y = 0, t > 0,
dt
with g(1) = 1.
Find g(t).
g(t) =
.
B. Let f (t) be the solution of the initial value problem
dy
5t + y = t 5
dt
with f (0) = 0.
Find f (t).
f (t) =
.
C. Find a constant c so that
20. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 18.pg
A Bernoulli differential equation is one of the form
dy
+ P(x)y = Q(x)yn .
dx
Observe that, if n = 0 or 1, the Bernoulli equation is linear.
For other values of n, the substitution u = y1−n transforms the
Bernoulli equation into the linear equation
du
+ (1 − n)P(x)u = (1 − n)Q(x).
dx
Use an appropriate substitution to solve the equation
xy0 + y = 2xy2 ,
k(t) = f (t) + cg(t)
and find the solution that satisfies y(1) = 5.
.
y(x) =
solves the differential equation in part B and k(1) = 14.
c=
.
17. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 10.pg
A. Let g(t) be the solution of the initial value problem
dy
+ 7y = 0,
dt
with y(0) = 1.
Find g(t).
g(t) =
.
B. Let f (t) be the solution of the initial value problem
dy
+ 7y = exp(1t)
dt
with y(0) = 1/8.
Find f (t).
f (t) =
.
C. Find a constant c so that
21. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 18a.pg
A Bernoulli differential equation is one of the form
dy
+ P(x)y = Q(x)yn (∗)
dx
Observe that, if n = 0 or 1, the Bernoulli equation is linear.
For other values of n, the substitution u = y1−n transforms the
Bernoulli equation into the linear equation
du
+ (1 − n)P(x)u = (1 − n)Q(x).
dx
Consider the initial value problem
xy0 + y = −3xy2 , y(1) = 4.
(a) This differential equation can be written in the form (∗) with
,
P(x) =
Q(x) =
, and
n=
.
(b) The substitution u =
will transform it into the linear
equation
du
+
u=
.
dx
(c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u:
.
u(1) =
(d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c).
u(x) =
.
(e) Finally, solve for y.
y(x) =
.
k(t) = f (t) + cg(t)
solves the differential equation in part B and k(0) = 12.
c=
.
18. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 12.pg
Find a family of solutions to the differential equation
(x2 − 4xy)dx + xdy = 0
(To enter the answer in the form below you may have to rearrange the equation so that the constant is by itself on one side of
the equation.) Then the solution in implicit form is:
the set of points (x, y) where F(x, y) =
= constant
2
Bernoulli equation into the linear equation
du
+ (1 − n)P(x)u = (1 − n)Q(x).
dx
Use an appropriate substitution to solve the equation
22. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 19.pg
A Bernoulli differential equation is one of the form
dy
+ P(x)y = Q(x)yn .
dx
9
y5
y0 − y = 8 ,
x
x
and find the solution that satisfies y(1) = 1.
y(x) =
.
Observe that, if n = 0 or 1, the Bernoulli equation is linear.
For other values of n, the substitution u = y1−n transforms the
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
3
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ5ModelingWith1stOrder due 01/05/2007 at 02:00am EST.
(c) As t becomes large, what value is y(t) approaching ? In other
(kg)
words, calculate the following limit. lim y(t) =
1.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 9.pg
t→∞
A curve passes through the point (0, 3) and has the property that
the slope of the curve at every point P is twice the y-coordinate
of P. What is the equation of the curve?
y(x) =
5.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ns7 4 31c.pg
A tank contains 100 kg of salt and 2000 L of water. A solution
of a concentration 0.025 kg of salt per liter enters a tank at the
rate 7 L/min. The solution is mixed and drains from the tank at
the same rate.
(a) What is the concentration of our solution in the tank initially?
(kg/L)
concentration =
(b) Find the amount of salt in the tank after 4.5 hours.
(kg)
amount =
(c) Find the concentration of salt in the solution in the tank as
time approaches infinity.
concentration =
(kg/L)
2.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ns7 4 31.pg
A tank contains 1240 L of pure water. A solution that contains
0.03 kg of sugar per liter enters a tank at the rate 9 L/min The
solution is mixed and drains from the tank at the same rate.
(a) How much sugar is in the tank initially?
(kg)
(b) Find the amount of sugar in the tank after t minutes.
amount =
(function of t)
(c) Find the concentration of sugar in the solution in the tank
after 90 minutes.
concentration =
6.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ns7 4 31d.pg
A tank contains 100 kg of salt and 1000 L of water. Pure water
enters a tank at the rate 12 L/min. The solution is mixed and
drains from the tank at the rate 6 L/min.
(a) What is the amount of salt in the tank initially?
amount =
(kg)
(b) Find the amount of salt in the tank after 2 hours.
amount =
(kg)
(c) Find the concentration of salt in the solution in the tank as
time approaches infinity. (Assume your tank is large enough to
hold all the solution.)
concentration =
(kg/L)
3.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ns7 4 31a.pg
A tank contains 2040 L of pure water. A solution that contains
0.06 kg of sugar per liter enters tank at the rate 7 L/min. The
solution is mixed and drains from the tank at the same rate.
(a) How much sugar is in the tank at the beginning.
y(0) =
(include units)
(b) With S representing the amount of sugar (in kg) at time t (in
minutes) write a differential equation which models this situation.
S0 = f (t, S) =
7.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ns7 4 31e.pg
.
Note: Make sure you use a capital S, (and don’t use S(t), it
confuses the computer). Don’t enter units for this function.
(c) Find the amount of sugar (in kg) after t minutes.
(function of t)
S(t) =
(d) Find the amout of the sugar after 84 minutes.
S(84) =
(include units)
Click here for help with units
A tank contains 1820 L of pure water. A solution that contains
0.07 kg of sugar per liter enters tank at the rate 9 L/min The
solution is mixed and drains from the tank at the same rate.
(a) How much sugar is in the tank at the beginning.
y(0) =
(include units)
(b) Find the amount of sugar (in kg) after t minutes.
y(t) =
(function of t)
(b) Find the amout of the sugar after 42 minutes.
y(42) =
(include units)
4.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ns7 4 31b.pg
/ns7 5 2.pg
A tank contains 1000 L of pure water. Solution that contains
0.03 kg of sugar per liter enters the tank at the rate 8 L/min, and
is thoroughly mixed into it. The new solution drains out of the
tank at the same rate.
(a) How much sugar is in the tank at the begining?
y(0) =
(kg)
(b) Find the amount of sugar after t minutes.
y(t) =
(kg)
A cell of some bacteria divides into two cells every 20 minutes.The initial population is 2 bacteria.
(a) Find the size of the population after t hours
y(t) =
(function of t)
(b) Find the size of the population after 4 hours.
y(4) =
(c) When will the population reach 8?
8.
1
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
T=
9.
determine the constant k, and then solve the equation to find an
expression for the size of the population after t years.
k=
,
P(t) =
.
(b) How long will it take for the population to increase to 4900
(half of the carrying capacity)?
It will take
years.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ur de 5 7.pg
A cell of some bacteria divides into two cells every 40 minutes.
The initial population is 200 bacteria.
(a) Find the population after t hours
(function of t)
y(t) =
(b) Find the population after 2 hours.
y(2) =
(c) When will the population reach 400?
T=
10.
14.
Another model for a growth function for a limited pupulation
is given by the Gompertz function, which is a solution of the
differential equation
dP
K
= c ln
P
dt
P
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ns7 5 3.pg
A bacteria culture starts with 160 bacteria and grows at a rate
proportional to its size. After 4 hours there will be 640 bacteria.
(a) Express the population after t hours as a function of t.
population:
(function of t)
(b) What will be the population after 7 hours?
where c is a constant and K is the carrying capacity.
(a) Solve this differential equation for c = 0.2, K = 5000, and
initial population P0 = 400.
.
P(t) =
(b) Compute the limiting value of the size of the population.
lim P(t) =
.
t→∞
(c) At what value of P does P grow fastest?
P=
.
(c) How long will it take for the population to reach 2930 ?
11.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ur de 5 12.pg
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ns7 6 1.pg
15.
A population P obeys the logistic model. It satisfies the equation
9
dP
=
P(11 − P) for P > 0.
dt
1100
(a) The population is increasing when
<P<
(b) The population is decreasing when P >
(c) Assume that P(0) = 4. Find P(84).
P(84) =
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ns7 5 10.pg
An unknown radioactive element decays into non-radioactive
substances. In 140 days the radioactivity of a sample decreases
by 77 percent.
(a) What is the half-life of the element?
half-life:
(days)
(b) How long will it take for a sample of 100 mg to decay to 41
mg?
time needed:
(days)
12.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 10.pg
Suppose that a population develops according to the logistic
equation
dP
= 0.25P − 0.0025P2
dt
where t is measured in weeks.
(a) What is the carriying capacity?
(b) Is the solution increasing or decreasing when P is between 0
and the carriying capacity? ?
(c) Is the solution increasing or decreasing when P is greater
than the carriying capacity? ?
16.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ur de 5 2.pg
A body of mass 6 kg is projected vertically upward with an initial velocity 61 meters per second.
The gravitational constant is g = 9.8m/s2 . The air resistance is
equal to k|v| where k is a constant.
Find a formula for the velocity at any time ( in terms of k ):
v(t) =
Find the limit of this velocity for a fixed time t0 as the air resistance coefficient k goes to 0. (Enter t0 as t 0 .)
v(t0 ) =
How does this compare with the solution to the equation for velocity when there is no air resistance?
This illustrates an important fact, related to the fundamental theorem of ODE and called ’continuous dependence’ on parameters and initial conditions. What this means is that, for a fixed
time, changing the initial conditions slightly, or changing the
parameters slightly, only slightly changes the value at time t.
The fact that the terminal time t under consideration is a fixed,
finite number is important. If you consider ’infinite’ t, or the
13.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 11.pg
Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species
in that lake) to be 9800. The number of fish doubled in the first
year.
(a) Assuming that the size of the fish population satisfies the
logistic equation
dP
P
= kP 1 −
,
dt
K
2
y0 = f (t, y) =
’final’ result you may get very different answers. Consider for
example a solution to y0 = y, whose initial condition is essentially zero, but which might vary a bit positive or negative. If
the initial condition is positive the ”final” result is plus infinity, but if the initial condition is negative the final condition is
negative infinity.
.
Note: Use y rather than y(t) since the latter confuses the computer. Don’t enter units for this equation.
Find an equation for the amount of money in the account at time
t where t is the number of years since January 1990.
y(t) =
Find the amount of money in the bank at the beginning of January 2000 (10 years later):
Find a solution to the equation which does not become infinite
(either positive or negative) over time:
y(t) =
During which months of the year does this non-growing solution
have the highest values? ?
17.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 8.pg
You have 600 dollars in your bank account. Suppose your
money is compounded every month at a rate of 0.5 percent per
month.
(a) How much do you have after t years?
(function of t)
y(t) =
(b) How much do you have after 110 months?
y(110) =
18.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
20.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-
/ur de 5 1.pg
/ur de 5 13.pg
A young person with no initial capital invests k dollars per year
in a retirement account at an annual rate of return 0.1. Assume
that investments are made continuously and that the return is
compounded continuously.
Determine a formula for the sum S(t) – (this will involve the
parameter k):
S(t) =
What value of k will provide 1589000 dollars in 47 years?
k=
How long will it take an investment to double in value if the
interest rate is 4% compounded continuously?
Answer:
years.
21.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 4.pg
Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its
temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of cooling. If
the coffee has a temperature of 210 degrees Fahrenheit when
freshly poured, and 1 minutes later has cooled to 199 degrees
in a room at 74 degrees, determine when the coffee reaches a
temperature of 159 degrees.
The coffee will reach a temperature of 159 degrees in
minutes.
19.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 3.pg
Here is a somewhat realistic example which combines the work
on earlier problems. You should use the phase plane plotter to
look at some solutions graphically before you start solving this
problem and to compare with your analytic answers to help you
find errors. You will probably be surprised to find how long it
takes to get all of the details of solution of a realistic problem
right, even when you know how to do each of the steps.
There are 1300 dollars in the bank account at the beginning of
January 1990, and money is added and withdrawn from the account at a rate which follows a sinusoidal pattern, peaking in
January and in July with money being added at a rate corresponding to 2110 dollars per year, while maximum withdrawals
take place at the rate of 490 dollars per year in April and October.
The interest rate remains constant at the rate of 6 percent per
year, compounded continuously.
Let y(t) represents the amount of money at time t (t is in years).
(dollars)
y(0) =
Write a formula for the rate of deposits and withdrawals (using
the functions sin(), cos() and constants):
g(t) =
The interest rate remains constant at 6 percent per year over this
period of time.
With y representing the amount of money in dollars at time t (in
years) write a differential equation which models this situation.
22.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 14.pg
A thermometer is taken from a room where the temperature is
25oC to the outdoors, where the temperature is 4oC. After one
minute the thermometer reads 19oC.
(a) What will the reading on the thermometer be after 5 more
minutes?
,
(b) When will the thermometer read 5oC?
minutes after it was taken to the outdoors.
23.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 5.pg
Susan finds an alien artifact in the desert, where there are temperature variations from a low in the 30s at night to a high in
the 100s in the day. She is interested in how the artifact will
respond to faster variations in temperature, so she kidnaps the
artifact, takes it back to her lab (hotly pursued by the military
police who patrol Area 51), and sticks it in an ”oven” – that is,
a closed box whose temperature she can control precisely.
3
Let T (t) be the temperature of the artifact. Newton’s law of
cooling says that T (t) changes at a rate proportional to the difference between the temperature of the environment and the
temperature of the artifact. This says that there is a constant
k, not dependent on time, such that T 0 = k(E − T ), where E is
the temperature of the environment (the oven).
Before collecting the artifact from the desert, Susan measured
its temperature at a couple of times, and she has determined that
for the alien artifact, k = 0.7.
Susan preheats her oven to 90 degrees Fahrenheit (she has stubbornly refused to join the metric world). At time t = 0 the oven
is at exactly 90 degrees and is heating up, and the oven runs
through a temperature cycle every 2π minutes, in which its temperature varies by 25 degrees above and 25 degrees below 90
degrees.
Let E(t) be the temperature of the oven after t minutes.
E(t) =
At time t = 0, when the artifact is at a temperature of 35 degrees, she puts it in the oven. Let T (t) be the temperature of the
(degrees)
artifact at time t. Then T (0) =
Write a differential equation which models the temperature of
the artifact.
T 0 = f (t, T ) =
y(0) =
(dollars)
With y representing the amount of money in dollars at time t (in
months) write a differential equation which models this situation.
y0 = f (t, y) =
.
Note: Use y rather than y(t) since the latter confuses the computer. Remember to check your units, but don’t enter units for
this equation – the computer won’t understand them.
0
Find an equation for the amount of money owed after t months.
y(t) =
Next we are going to think about how many months it will take
until the loan is paid off. Remember that y(t) is the amount that
is owed after t months. The loan is paid off when y(t) =
Once you have calculated how many months it will take to
pay off the loan, give your answer as a decimal, ignoring
the fact that in real life there would be a whole number of
months. To do this, you should use a graphing calculator or
use a plotter on this page to estimate the root. If you use the
the xFunctions plotter, then once you have launched xFunctions, pull down the Multigaph Utility from the menu in the upper right hand corner, enter the function you got for y (using x
as the independent variable, sorry!), choose appropriate ranges
for the axes, and then eyeball a solution.
The loan will be paid off in
months.
If the borrower wanted the loan to be paid off in exactly 20
years, with the same payment plan as above, how much could
be borrowed?
Borrowed amount =
.
Note: Use T rather than T (t) since the latter confuses the computer. Don’t enter units for this equation.
Solve the differential equation. To do this, you may find it helpful to know that if a is a constant, then
Z
1
sin(t)eat dt = 2
eat (a sin(t) − cos(t)) +C.
a +1
T (t) =
After Susan puts in the artifact in the oven, the military police
break in and take her away. Think about what happens to her
artifact as t → ∞ and fill in the following sentence:
For large values of t, even though the oven temperature varies
between 65 and 115 degrees, the artifact varies from
to
degrees.
25.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 15.pg
24.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 6.pg
Here is a multipart example on finance. Be patient and careful
as you work on this problem. You will probably be surprised to
find how long it takes to get all of the details of the solution of a
realistic problem exactly right, even when you know how to do
each of the steps. Use the computer to check the steps for you
as you go along. There is partial credit on this problem.
A recent college graduate borrows 100000 dollars at an (annual) interest rate of 9 per cent. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate
of 600(1 + t/160) dollars per month, where t is the number of
months since the loan was made.
Let y(t) be the amount of money that the graduate owes t months
after the loan is made.
In the circuit shown in the figure above a battery supplies a constant voltage of E = 40V, the inductance is L = 2H, the resistance is R = 30 Ω, and I(0) = 0. Find the current after t seconds.
I(t) =
.
4
the capacitor is Q/C, where Q is the charge (in coulombs), so in
this case Kirchhoff’s Law gives
26.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 16.pg
RI +
Since I =
Q
= E(t).
C
dQ
, we have
dt
R
dQ 1
+ Q = E(t).
dt
C
Suppose the resistance is 20 Ω, the capacitance is 0.2F, a battery gives a constant voltage of 50V, and the initial charge is
Q(0) = 0C.
Find the charge and the current at time t.
,
Q(t) =
I(t) =
.
In the circuit shown in the figure above a generator supplies a
voltage of E(t) = 60 sin(50t)V, the inductance is L = 2H, the
resistance is R = 10 Ω, and I(0) = 1. Find the current after t
seconds.
I(t) =
.
28.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 18.pg
Let P(t) be the performance level of someone learning a skill
dP
as a function of the training time t. The derivative
repredt
sents the rate at which performance improves. If M is the maximum level of performance of which the learner is capable, then
a model for learning is given by the differential equation
27.
(1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder/ur de 5 17.pg
dP
= k(M − P(t))
dt
where k is a positive constant.
Two new workers, Andy and Bob, were hired for an assembly
line. Andy could process 12 units per minute after one hour
and 13 units per minute after two hours. Bob could process 11
units per minute after one hour and 14 units per minute after two
hours. Using the above model and assuming that P(0) = 0, estimate the maximum number of units per minute that each worker
is capable of processing.
Andy:
,
Bob:
.
The figure above shows a circuit containing an electromotive
force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms Ω. The voltage drop across
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
5
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ6AutonomousStability due 01/06/2007 at 02:00am EST.
1.
(1
pt)
Given the differential equation x0 (t) = f (x(t)).
List the constant (or equilibrium) solutions to this differential
equation in increasing order and indicate whether or not these
equations are stable, semi-stable, or unstable.
?
?
?
?
rochesterLibrary/setDiffEQ6AutonomousStability-
/ur de 6 1.pg
The graph of the function f (x) is
3.
(1
pt)
rochesterLibrary/setDiffEQ6AutonomousStability-
/ur de 6 3.pg
Given the differential equation x0 = −(x + 3.5) ∗ (x + 2)3 (x +
0.5)2 (x − 1).
List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not
these equations are stable, semi-stable, or unstable. (It helps
to sketch the graph. xFunctions will plot functions as well as
phase planes. )
?
?
?
?
(the horizontal axis is x.)
Consider the differential equation x0 (t) = f (x(t)).
List the constant (or equilibrium) solutions to this differential
equation in increasing order and indicate whether or not these
equations are stable, semi-stable, or unstable.
?
?
?
?
2.
(1
pt)
4.
pt)
rochesterLibrary/setDiffEQ6AutonomousStability-
Given the differential equation x0 (t) = x4 + 2x3 − 5.5x2 − 0.5x +
1.3125.
List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not
these equations are stable, semi-stable, or unstable. (It helps
to sketch the graph. xFunctions will plot functions as well as
phase planes. )
?
?
?
?
rochesterLibrary/setDiffEQ6AutonomousStability-
/ur de 6 2.pg
The graph of the function f (x) is
(the horizontal axis is x.)
(1
/ur de 6 4.pg
1
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ7Exact due 01/07/2007 at 02:00am EST.
?
F(x, y) =
1. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 1.pg
The following differential equation is exact.
Find a function F(x,y) whose level curves are solutions to the
differential equation
5. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 5.pg
Use the ”mixed partials” check to see if the following differential equation is exact.
If it is exact find a function F(x,y) whose level curves are solutions to the differential equation
ydy − xdx = 0
F(x, y) =
2. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 2.pg
Use the ”mixed partials” check to see if the following differential equation is exact.
If it is exact find a function F(x,y) whose level curves are solutions to the differential equation
4
(3ex sin(y) − 4y)dx + (−4x + 3ex cos(y))dy = 0
?
F(x, y) =
4
(4x + 2y)dx + (4x + 3y )dy = 0
?
F(x, y) =
6. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 6.pg
Check that the equation below is not exact but becomes exact
when multiplied by the integrating factor.
3. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 3.pg
Use the ”mixed partials” check to see if the following differential equation is exact.
If it is exact find a function F(x,y) whose level curves are solutions to the differential equation
x2 y3 + x(1 + y2 )y0 = 0
Integrating factor: µ(x, y) = 1/(xy3 ).
Solve the differential equation.
You can define the solution curve implicitly by a function in the
form
F(x, y) = G(x) + H(y) = K F(x, y) =
(−1xy2 − 3y)dx + (−1x2 y − 3x)dy = 0
?
F(x, y) =
4. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 4.pg
Use the ”mixed partials” check to see if the following differential equation is exact.
If it is exact find a function F(x,y) whose level curves are solutions to the differential equation
7. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 7.pg
Find an explicit or implicit solutions to the differential equation
(x2 + 2xy)dx + xdy = 0
dy +4x2 + 2y
=
dx −2x + 4y1
F(x, y) =
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
1
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ8FundTheorem due 01/08/2007 at 02:00am EST.
1. (1 pt) rochesterLibrary/setDiffEQ8FundTheorem/ur de 8 1.pg
This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can’t be
graded by WeBWorK, but is to be handed in at the first class after the due date.
A. State the uniqueness property of the fundamental theorem.
B. Show directly using the differential equation, that if y1 (t) is a solution to the differential equation y0 (t) = y(t), then y2 (t) =
y1 (t + a) is also a solution to the differential equation. (You will need to use the known facts about y1 to calculate that y02 (t) = y2 (t)
). (We know that the solution is the exponential function, but you will not need to use this fact.)
C. Describe the relationship between the graphs of y1 and y2 and using a sketch of the direction field explain why it is obvious
that if y1 is a solution then y2 has to be a solution also.
D. Describe in words why if y1 (t) is any solution to the differential equation y0 = f (y) then y2 (t) = y1 (t + a) is also a solution.
E. Show that if y1 (t) solves y0 (t) = y(t), then y2 (t) = Ay1 (t) also solves the same equation.
F. Suppose that y1 (t) solves y0 (t) = y(t) and y(0) = 1. (Such a solution is guaranteed by the fundamental theorem.). Let y2 (t) =
y1 (t + a) and let y3 (t) = y1 (a)y1 (t). Calculate the values y2 (0) and y3 (0). Use the uniqueness property to show that y2 (t) = y3 (t) for
all t.
G. Explain how this proves that any solution to y0 = y must be a function which obeys the law of exponents.
H. Let z = x + iy. Define exp(z) ( or ez ) using a Taylor series. Show that if z = x + iy is a constant, then
d
exp(tz) = z exp(tz)
dt
by differentiating the power series.
I. Use your earlier results to show that exp(z + w) = exp(z) exp(w). This method of checking the law of exponents is MUCH
easier than expanding the power series.
You can find a direction field plotter here or at the direction field plotter page. Choose ”integral curves utility” from the ”main
screen” menu of xFunctions to get to the phaseplane plotter.
2. (1 pt) rochesterLibrary/setDiffEQ8FundTheorem/ur de 8 2.pg
This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can’t be
graded by WeBWorK, but is to be handed in at the first class after the due date.
A. Using the same technique as in the previous problem show that if a function y1 (t) satisfies: (1) y1 (0) = 1 and (2) y0 (t) = y(t)
then
(y1 (t))r = y1 (rt)
B. Explain in words how this relates to another law of exponents.
You can find a direction field plotter here or at the direction field plotter page. Choose ”integral curves utility” from the ”main
screen” menu of xFunctions to get to the phaseplane plotter.
1
3. (1 pt) rochesterLibrary/setDiffEQ8FundTheorem/ur de 8 3.pg
This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can’t be
graded by WeBWorK, but is to be handed in at the first class after the due date.
Use the same ideas as in the previous problems.
A. Suppose that y1 (t) satisfies the equation y00 + y = 0 and y1 (0) = 0and y01 (0) = 1. Such a function exists because of the
fundamental theorem. (We all know that it is sin(t), but you should not use that fact in answering the questions below.)
Show that y2 (t) = y01 (t) also satisfies the equation y00 + y = 0 and that y2 (0) = 1 and y02 (0) = 0.
B. If y3 (t) = y02 (t) show, using the uniqueness property, that y3 (t) = −y1 (t)
C. State the uniqueness property for solutions to second order differential equations (or equivalently to a system of two first order
differential equations).
D. Use the uniqueness property to show that y1 (t + a) = y01 (a)y1 (t) + y1 (a)y2 (t) = y2 (a)y1 (t) + y1 (a)y2 (t)
The formulas for the sin of sums of angles can be calculated completely from the one fact that it satisfies a differential equation.
This is a general fact. Any solution of a differential equation has the potential for obeying certain ”laws” which are dictated by the
differential equation.
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2
Arnie Pizer
Rochester Problem Library Fall 2005
WeBWorK assignment DiffEQ9Linear2ndOrderHomog due 01/09/2007 at 02:00am EST.
1.
7.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
Find y as a function of t if
36y00 + 72y0 + 56y = 0,
Find y as a function of t if
00
4y − 729y = 0,
2.
y(0) = 7, y0 (0) = 2.
y=
Note: This problem cannot interpret complex numbers. You
may need to simplify your answer before submitting it.
y0 (0) = 2.
y(0) = 7,
y(t) =
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 6.pg
8.
Find y as a function of t if
y(0) = 3,
Find y as a function of t if
250y00 + 16y0 + 9y = 0,
y0 (0) = 6.
y(0) = 7, y0 (0) = 2.
y(t) =
Note: This problem cannot interpret complex numbers. You
may need to simplify your answer before submitting it.
3.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 8.pg
9.
Find y as a function of t if
4.
Find y as a function of t if
y00 + 6y0 + 25y = 0,
y0 (0) = 4.
/ur de 9 2.pg
Find y as a function of t if
y0 (0) = 4.
10.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 9.pg
y00 − 9y0 = 0,
Find y as a function of t if
81y00 + 18y0 = 0,
y(0) = 3, y(1) = 3.
y(t) =
Remark: The initial conditions involve values at two points.
y(0) = 1, y0 (0) = 2.
y=
Note: This problem cannot interpret complex numbers. You
may need to simplify your answer before submitting it.
5.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 7.pg
Find y as a function of t if
11.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 10.pg
36y00 − 12y0 + y = 0,
6.
y(0) = 4,
y=
Note: This problem cannot interpret complex numbers. You
may need to simplify your answer before submitting it.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
Find the function y1 of t which is the solution of
y0 (0) = 3.
y(0) = 7,
y=
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 11.pg
10000y00 + 729y = 0,
y(0) = 7,
y=
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 5.pg
10000y00 − 9y = 0
with
y=
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 4.pg
/ur de 9 1.pg
100y00 + 160y0 + 15y = 0
with initial conditions y1 (0) = 1, y01 (0) = 0.
y1 =
Find the function y2 of t which is the solution of
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 3.pg
Find y as a function of t if
100y00 + 160y0 + 15y = 0
6y00 + 32y = 0,
y(0) = 7, y0 (0) = 2.
y(t) =
Note: This particular weBWorK problem can’t handle complex
numbers, so write your answer in terms of sines and cosines,
rather than using e to a complex power.
with initial conditions
y2 =
Find the Wronskian
y2 (0) = 0,
y02 (0) = 1.
W (t) = W (y1 , y2 ).
W (t) =
1
Remark: You can find W by direct computation and use Abel’s
theorem as a check. You should find that W is not zero and so
y1 and y2 form a fundamental set of solutions of
? 4. The equations
a f1 (2) + b f2 (2) = c
a f10 (2) + b f20 (2) = d
100y00 + 160y0 + 15y = 0.
12.
have a unique solution for any c and d
? 5. The vectors ( f1 (0), f10 (0)) and ( f2 (0), f20 (0)) are linearly independent
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 12.pg
Find y as a function of t if
1296y00 − 360y0 + 25y = 0,
0
y(0) = 2, y (0) = 4.
y=
16.
Determine which of the following pairs of functions are linearly
independent.
13.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 13.pg
?
?
?
?
Find y as a function of t if
100y00 + 160y0 + 100y = 0,
y(3) = 4, y0 (3) = 9.
y=
1.
2.
3.
4.
f (t) = eλt cos(µt) , g(t) = eλt sin(µt) , µ 6= 0
f (θ) = cos(3θ) , g(θ) = 2 cos3 (θ) − 4 cos(θ)
f (t) = 2t 2 + 14t , g(t) = 2t 2 − 14t
f (x) = x2 , g(x) = 4|x|2
17.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 17.pg
14.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 14.pg
Match the second order linear equations with the Wronskian of
(one of) their fundamental solution sets.
1. y00 − ln(t)y0 − 9y = 0
2. y00 + 4y0 − 9y = 0
3. y00 − 1t y0 − 9y = 0, t > 0
4. y00 − 2t y0 − 9y = 0
5. y00 + 2t y0 − 9y = 0
Determine whether the following pairs of functions are linearly
independent or not.
? 1. f (t) = t 2 + 9t and g(t) = t 2 − 9t
? 2. The Wronskian of two functions is W (t) = t are the
functions linearly independent or dependent?
? 3. f (t) = t and g(t) = |t|
15.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 16.pg
A.
B.
C.
D.
E.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 15.pg
Suppose that the Wronskian of two functions f1 (t) and f2 (t) is
f (t) f2 (t)
given by W (t) = t 2 − 4 = det 10
Even though you
f1 (t) f20 (t)
don’t know the functions f1 and f2 you can determine whether
the following questions are true or false.
18.
W (t) = 2e−4t
W (t) = t 2
W (t) = et ln(t)−t
W (t) = t52
W (t) = 7t
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-
/ur de 9 18.pg
? 1. The vectors ( f1 (−2), f10 (−2)) and ( f2 (−2), f20 (−2))
are linearly independent
? 2. The equations
Find y as a function of x if
x2 y00 + 2xy0 − 12y = 0,
y(1) = −4, y0 (1) = −10.
y=
a f1 (2) + b f2 (2) = 0
a f10 (2) + b f20 (2) = 0
19.
(1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog/ur de 9 19.pg
have more than one solution.
? 3. The equations
Find y as a function of x if
x2 y00 − 7xy0 + 16y = 0,
a f1 (0) + b f2 (0) = c
a f10 (0) + b f20 (0) = d
y(1) = 5, y0 (1) = 3.
y=
have a unique solution for any c and d
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
2