Exam 1 Review
Show that y = y(x) is/is not a solution of the differential equation F (x, y, y 0, . . . y (n)) = 0.
Examples:
1. x2 y 00 − 3x y 0 + 4y = 0;
2.
d3y
dy
+
= ex ;
3
dx
dx
3. xy 00 + y 0 = 0;
y1 (x) = x2 , y2 (x) = x2 ln x
y(x) = 1 + sin x + 12 ex , z(x) = 2 cos x + 12 ex .
y1 (x) = ln (1/x), y2 (x) = x2.
4. (x + 1)y 00 + xy 0 − y = (x + 1)2;
5. y 0 + y = y 2 ;
y=
y(x) = e−x + x2 + 1, z(x) = x2 + 1.
1
.
Cex + 1
Find the differential equation for an n-parameter family of curves.
Examples:
1. y 2 = Cx3 − 2.
2. y = C1 x3 +
C2
.
x
3. y = C1 e−2x + C2 xe−2x .
4. y 3 = C(x − 2)2 + 4
Identify each of the following first order differential equations.
1. x(1 + y 2) + y(1 + x2 )y 0 = 0.
2. (xy + y)y 0 = x − xy.
3. xy 2
dy
= x3 ey/x − x2 y
dx
4. y 0 = −
3y
+ x4 y 1/3.
x
5. (3x2 + 1)y 0 − 2xy = 6x.
6. x(1 − y) + y(1 + x2)
dy
= 0.
dx
7. xy 0 = x2y + y 2 ln x.
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First order linear equations; find general solution, solve an initial-value problem.
Examples:
1. Find the general solution of x2 dy − 2xy dx = x4 cos 2x dx.
2. Find the general solution of (1 + x2 ) y 0 + 1 + 2x y = 0.
3. Find the general solution of xy 0 − y = 2x ln x.
4. Find the solution of the initial-value problem xy 0 + 3y =
ex
,
x
y(1) = 2.
Separable equations; find general solution, solve an initial-value problem.
Examples:
1. Find the general solution of y 0 = xex+y .
2. Find the general solution of yy 0 = xy 2 − x − y 2 + 1.
3. Find the general solution of ln x
dy
y
= .
dx
x
4. Find the solution of the initial-value problem y 0 =
x2 y − y
,
y+1
y(3) = 1.
Bernoulli equations; find general solution.
Examples:
1. Find the general solution of y 0 + xy = xy 3 .
√
2. Find the general solution of y 0 = 4y + 2ex y.
Homogeneous equations; find general solution.
Examples:
1. Find the general solution of y 0 =
y2
.
xy + y 2
2. Find the general solution of y 0 =
x2ey/x + y 2
.
xy
Applications
Examples:
1. Given the family of curves y = Ce2x + 1. Find the family of orthogonal trajectories.
2. Given the family of curves y 2 = C(x + 2)3 − 2. Find the family of orthogonal trajectories.
3. A 200 gallon tank, initially full of water, develops a leak at the bottom. Given that 20% of
the water leaks out in the first 4 minutes, find the amount of water left in the tank t minutes
after the leak develops if:
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(i) The water drains off a rate proportional to the amount of water present.
(ii) The water drains off a rate proportional to the product of the time elapsed and the
amount of water present.
(iii) The water drains off a rate proportional to the square root of the amount of water
present.
4. A certain radioactive material is decaying at a rate proportional to the amount present. If a
sample of 100 grams of the material was present initially and after 2 hours the sample lost
20% of its mass, find:
(a) An expression for the mass of the material remaining at any time t.
(b) The mass of the material after 4 hours.
(c) The half-life of the material.
5. A biologist observes that a certain bacterial colony triples every 4 hours and after 12 hours
occupies 1 square centimeter. Assume that the colony obeys the population growth law.
(a) How much area did the colony occupy when first observed?
(b) What is the doubling time for the colony?
6. A thermometer is taken from a room where the temperature is 72o F to the outside where the
temperature is 32o F . After 1/2 minute, the thermometer reads 50o F . Assume Newton’s
Law of Cooling.
(a) What will the thermometer read after it has been outside for 1 minute?
(b) How many minutes does the thermometer have to be outside for it to read 35o F ?
7. An advertising company designs a campaign to introduce a new product to a metropolitan
area of population M . Let P = P (t) denote the number of people who become aware of
the product by time t. Suppose that P increases at a rate proportional to the number of
people still unaware of the product. The company determines that no one is aware of the
product at the beginning of the campaign [P (0) = 0] and that 30% of the people were
aware of the product after 10 days of advertising.
(a) Give the differential equation that describes the number of people who become aware of
the product by time t.
(b) Determine the solution of the differential equation in (a) that satisfies the initial condition
P (0) = 0.
(c) How long does it take for 90% of the population to become aware of the product?
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Second order linear equations; general theory.
Examples:
1. The functions y1 (x) = 2x4 − x−2 , y2 (x) = 3x−2 , y3 (x) = x4 + 2x−2 are solutions of
x2y 00 − xy 0 − 8y = 0.
(i) Find a solution basis for the equation and justify your answer.
(ii) Give the general solution of the equation.
2. Given the differential equation x2y 00 − 2x y 0 − 10 y = 0.
(a) Find two values of r such that y = xr is a solution of the equation.
(b) Determine a fundamental set of solutions and give the general solution of the equation.
(c) Find the solution of the equation satisfying the initial conditions y(1) = 6, y 0 (1) = 2.
00
3. Given the differential equation y −
(a)
(b)
(c)
(d)
6 0
y +
x
12
y = 0.
x2
Find two values of r such that y = xr is a solution of the equation.
Determine a fundamental set of solutions and give the general solution of the equation.
Find the solution of the equation satisfying the initial conditions y(1) = 2, y 0 (1) = −1.
Find the solution of the equation satisfying the initial conditions y(2) = y 0 (2) = 0.
Homogeneous equations with constant coefficients.
Examples:
1. Find the general solution of y 00 + 10y 0 + 25 y = 0.
2. Find the general solution of y 00 − 8 y 0 + 15 y = 0.
3. Find the general solution of y 00 + 4y 0 + 20y = 0.
4. Find the solution of the initial-value problem: y 00 − 2y 0 + 2y = 0;
y(0) = −1, y 0(0) = −1.
5. Find the solution of the initial-value problem: y 00 + 4y 0 + 4y = 0;
y(−1) = 2, y 0(−1) = 1.
6. Find the general solution of y 00 − 2α y 0 + α2 y = 0,
α a constant.
7. Find the general solution of y 00 − 2α y 0 + (α2 + β 2 )y = 0,
α, β constants.
8. The function y = 2e3x − 5e−4x is a solution of a second order linear differential equation
with constant coefficients. What is the equation?
9. The function y = e−3x cos 2x is a solution of a second order linear differential equation with
constant coefficients. What is the equation?
10. Find a second order linear homogeneous differential equation with constant coefficients that
has y = e−4x as a solution.
11. The function y = x e4x is a solution of a second order linear differential equation with
constant coefficients. What is the equation?
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