Zill, D., Cullen, M. Differential equations with boundary-value problems
50
CHAPTER 2
●
FIRST-ORDER DIFFERENTIAL EQUATIONS
REMARKS
(i) As we have just seen in Example 5, some simple functions do not possess
an antiderivative that is an elementary function. Integrals of these kinds of
2
functions are called nonelementary. For example, x3 et dt and sin x2 dx are
nonelementary integrals. We will run into this concept again in Section 2.3.
(ii) In some of the preceding examples we saw that the constant in the oneparameter family of solutions for a first-order differential equation can be relabeled when convenient. Also, it can easily happen that two individuals solving the
same equation correctly arrive at dissimilar expressions for their answers. For
example, by separation of variables we can show that one-parameter families of
solutions for the DE (1 y 2 ) dx (1 x 2 ) dy 0 are
arctan x arctan y c
or
xy
c.
1 xy
As you work your way through the next several sections, bear in mind that families of solutions may be equivalent in the sense that one family may be obtained
from another by either relabeling the constant or applying algebra and trigonometry. See Problems 27 and 28 in Exercises 2.2.
Separáveis
EXERCISES 2.2
In Problems 1–22 solve the given differential equation by
separation of variables.
1.
dy
sin 5x
dx
2.
3. dx e 3xdy 0
5. x
7.
6.
dy
e3x2y
dx
dy
2xy 2 0
dx
8. e x y
y1
dx
dy
x
2
10.
dy
ey e2xy
dx
2y 3
dy
dx
4x 5
dx y(1 x )
dS
kS
dr
dP
17.
P P2
dt
dx
4(x2 1), x(>4) 1
dt
24.
dy y2 1
,
dx x2 1
dy
y2
dx
25. x2
26.
y(2) 2
dy
y xy, y(1) 1
dx
dy
2y 1,
dt
y(0) 52
28. (1 x 4 ) dy x(1 4y 2 ) dx 0,
13. (e y 1) 2ey dx (e x 1) 3ex dy 0
15.
22. (ex ex )
27. 11 y2 dx 11 x2 dy 0, y(0) 12. sin 3x dx 2y cos 33x dy 0
14. x(1 y )
dy
x11 y2
dx
23.
2
11. csc y dx sec 2x dy 0
2 1/2
21.
In Problems 23 –28 find an explicit solution of the given
initial-value problem.
4. dy (y 1) 2 dx 0
dy
4y
dx
9. y ln x
dy
(x 1)2
dx
Answers to selected odd-numbered problems begin on page ANS-1.
2 1/2
dy
29.
dy
2
yex , y(4) 1
dx
dN
18.
N Ntet2
dt
30.
dy
y 2 sin x2,
dx
dy xy 2y x 2
dy
xy 3x y 3
19.
20.
dx xy 2x 4y 8
dx xy 3y x 3
y(1) 0
In Problems 29 and 30 proceed as in Example 5 and find an
explicit solution of the given initial-value problem.
dQ
k(Q 70)
dt
16.
13
2
y(2) 13
31. (a) Find a solution of the initial-value problem consisting
of the differential equation in Example 3 and the initial conditions y(0) 2, y(0) 2, and y 14 1.
()
2.2
(b) Find the solution of the differential equation in
Example 4 when ln c1 is used as the constant of
integration on the left-hand side in the solution and
4 ln c1 is replaced by ln c. Then solve the same
initial-value problems in part (a).
dy
32. Find a solution of x
y2 y that passes through
dx
the indicated points.
(a) (0, 1)
(b) (0, 0)
(c) 12, 12
(d) 2, 14
( )
( )
33. Find a singular solution of Problem 21. Of Problem 22.
34. Show that an implicit solution of
2x sin2 y dx (x 2 10) cos y dy 0
is given by ln(x 2 10) csc y c. Find the constant
solutions, if any, that were lost in the solution of the differential equation.
Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the
initial condition or the equation itself. In Problems 35– 38 find
an explicit solution of the given initial-value problem. Use a
graphing utility to plot the graph of each solution. Compare
each solution curve in a neighborhood of (0, 1).
35.
dy
(y 1)2,
dx
y(0) 1
dy
(y 1)2, y(0) 1.01
dx
dy
37.
(y 1)2 0.01, y(0) 1
dx
36.
38.
dy
(y 1)2 0.01,
dx
y(0) 1
39. Every autonomous first-order equation dydx f (y)
is separable. Find explicit solutions y1(x), y 2(x), y 3(x),
and y 4(x) of the differential equation dydx y y 3
that satisfy, in turn, the initial conditions y1(0) 2,
y2(0) 12, y3(0) 12, and y 4(0) 2. Use a graphing
utility to plot the graphs of each solution. Compare these
graphs with those predicted in Problem 19 of Exercises
2.1. Give the exact interval of definition for each solution.
40. (a) The autonomous first-order differential equation
dydx 1( y 3) has no critical points.
Nevertheless, place 3 on the phase line and obtain
a phase portrait of the equation. Compute d 2 ydx 2
to determine where solution curves are concave up
and where they are concave down (see Problems 35
and 36 in Exercises 2.1). Use the phase portrait
and concavity to sketch, by hand, some typical
solution curves.
(b) Find explicit solutions y1(x), y 2(x), y 3(x), and y 4(x)
of the differential equation in part (a) that satisfy,
in turn, the initial conditions y1(0) 4, y 2(0) 2,
SEPARABLE VARIABLES
●
51
y3(1) 2, and y 4(1) 4. Graph each solution
and compare with your sketches in part (a). Give
the exact interval of definition for each solution.
41. (a) Find an explicit solution of the initial-value problem
dy 2x 1
, y(2) 1.
dx
2y
(b) Use a graphing utility to plot the graph of the solution in part (a). Use the graph to estimate the interval I of definition of the solution.
(c) Determine the exact interval I of definition by analytical methods.
42. Repeat parts (a) – (c) of Problem 41 for the IVP consisting of the differential equation in Problem 7 and the initial condition y(0) 0.
Discussion Problems
43. (a) Explain why the interval of definition of the explicit
solution y ␾ 2 (x) of the initial-value problem in
Example 2 is the open interval (5, 5).
(b) Can any solution of the differential equation cross
the x-axis? Do you think that x 2 y 2 1 is an
implicit solution of the initial-value problem
dydx xy, y(1) 0?
44. (a) If a 0, discuss the differences, if any, between
the solutions of the initial-value problems consisting of the differential equation dydx xy and
each of the initial conditions y(a) a, y(a) a,
y(a) a, and y(a) a.
(b) Does the initial-value problem dydx xy,
y(0) 0 have a solution?
(c) Solve dydx xy, y(1) 2 and give the exact
interval I of definition of its solution.
45. In Problems 39 and 40 we saw that every autonomous
first-order differential equation dydx f (y) is
separable. Does this fact help in the solution of the
dy
initial-value problem
11 y2 sin2 y, y(0) 12?
dx
Discuss. Sketch, by hand, a plausible solution curve of
the problem.
46. Without the use of technology, how would you solve
dy
1y y?
( 1x x) dx
Carry out your ideas.
47. Find a function whose square plus the square of its
derivative is 1.
48. (a) The differential equation in Problem 27 is equivalent to the normal form
dy
1 y2
dx
B1 x 2
60
●
CHAPTER 2
FIRST-ORDER DIFFERENTIAL EQUATIONS
USE OF COMPUTERS The computer algebra systems Mathematica and Maple
are capable of producing implicit or explicit solutions for some kinds of differential
equations using their dsolve commands.*
REMARKS
(i) In general, a linear DE of any order is said to be homogeneous when
g(x) 0 in (6) of Section 1.1. For example, the linear second-order DE
y 2y 6y 0 is homogeneous. As can be seen in this example and in the
special case (3) of this section, the trivial solution y 0 is always a solution of
a homogeneous linear DE.
(ii) Occasionally, a first-order differential equation is not linear in one variable
but is linear in the other variable. For example, the differential equation
dy
1
dx x y 2
is not linear in the variable y. But its reciprocal
dx
x y2
dy
or
dx
x y2
dy
is recognized as linear in the variable x. You should verify that the integrating
factor e (1)dy ey and integration by parts yield the explicit solution
x y 2 2y 2 ce y for the second equation. This expression is, then,
an implicit solution of the first equation.
(iii) Mathematicians have adopted as their own certain words from engineering, which they found appropriately descriptive. The word transient, used
earlier, is one of these terms. In future discussions the words input and output
will occasionally pop up. The function f in (2) is called the input or driving
function; a solution y(x) of the differential equation for a given input is called
the output or response.
(iv) The term special functions mentioned in conjunction with the error function also applies to the sine integral function and the Fresnel sine integral
introduced in Problems 49 and 50 in Exercises 2.3. “Special Functions” is
actually a well-defined branch of mathematics. More special functions are
studied in Section 6.3.
*Certain commands have the same spelling, but in Mathematica commands begin with a capital letter
(Dsolve), whereas in Maple the same command begins with a lower case letter (dsolve). When
discussing such common syntax, we compromise and write, for example, dsolve. See the Student
Resource and Solutions Manual for the complete input commands used to solve a linear first-order DE.
EXERCISES 2.3
Lineares
In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the
general solution is defined. Determine whether there are any
transient terms in the general solution.
1.
dy
5y
dx
dy
3.
y e3x
dx
2.
dy
2y 0
dx
dy
4. 3
12y 4
dx
Answers to selected odd-numbered problems begin on page ANS-2.
5. y 3x 2 y x 2
6. y 2xy x 3
7. x 2 y xy 1
8. y 2y x 2 5
9. x
dy
y x 2 sin x
dx
10. x
11. x
dy
4y x 3 x
dx
12. (1 x)
13. x 2y x(x 2)y e x
dy
2y 3
dx
dy
xy x x 2
dx
2.3
14. xy (1 x)y ex sin 2x
15. y dx 4(x y 6) dy 0
33.
f (x) dy
(sin x)y 1
dx
dy
18. cos2x sin x
(cos3x)y 1
dx
dy
(x 2)y 2xex
dx
dy
20. (x 2)2
5 8y 4xy
dx
dr
21.
r sec cos d
dP
22.
2tP P 4t 2
dt
19. (x 1)
23. x
dy
(3x 1)y e3x
dx
24. (x 2 1)
dy
2y (x 1)2
dx
In Problems 25 – 30 solve the given initial-value problem.
Give the largest interval I over which the solution is defined.
25. xy y ex,
26. y
y(1) 2
●
61
dy
2xy f (x), y(0) 2, where
dx
16. y dx (ye y 2x) dy
17. cos x
LINEAR EQUATIONS
34. (1 x 2)
x,0,
0x1
x1
dy
2xy f (x), y(0) 0, where
dx
f (x) x,x,
0x1
x1
35. Proceed in a manner analogous to Example 6 to solve the
initial-value problem y P(x)y 4x, y(0) 3, where
P(x) 2,
2>x,
0 x 1,
x 1.
Use a graphing utility to graph the continuous function
y(x).
36. Consider the initial-value problem y e x y f (x),
y(0) 1. Express the solution of the IVP for x 0 as a
nonelementary integral when f (x) 1. What is the solution when f (x) 0? When f (x) e x?
37. Express the solution of the initial-value problem
y 2xy 1, y(1) 1, in terms of erf(x).
Discussion Problems
dx
x 2y2, y(1) 5
dy
di
Ri E, i(0) i0,
dt
L, R, E, and i 0 constants
27. L
38. Reread the discussion following Example 2. Construct a
linear first-order differential equation for which all
nonconstant solutions approach the horizontal asymptote y 4 as x : .
y(1) 10
39. Reread Example 3 and then discuss, with reference
to Theorem 1.2.1, the existence and uniqueness of a
solution of the initial-value problem consisting of
xy 4y x 6e x and the given initial condition.
(a) y(0) 0
(b) y(0) y 0 , y 0 0
(c) y(x 0) y 0 , x 0 0, y 0 0
30. y (tan x)y cos 2 x, y(0) 1
40. Reread Example 4 and then find the general solution of
the differential equation on the interval (3, 3).
In Problems 31 – 34 proceed as in Example 6 to solve the
given initial-value problem. Use a graphing utility to graph
the continuous function y(x).
41. Reread the discussion following Example 5. Construct a
linear first-order differential equation for which all solutions are asymptotic to the line y 3x 5 as x : .
28.
dT
k(T Tm ); T(0) T0,
dt
k, T m, and T 0 constants
29. (x 1)
31.
dy
y ln x,
dx
dy
2y f (x), y(0) 0, where
dx
f (x) 32.
1,0,
0x3
x3
dy
y f (x), y(0) 1, where
dx
f (x) 1,1,
0x1
x1
42. Reread Example 6 and then discuss why it is technically
incorrect to say that the function in (13) is a “solution”
of the IVP on the interval [0, ).
43. (a) Construct a linear first-order differential equation of
the form xy a 0 (x)y g(x) for which yc cx 3
and yp x 3. Give an interval on which
y x 3 cx 3 is the general solution of the DE.
(b) Give an initial condition y(x 0) y 0 for the DE
found in part (a) so that the solution of the IVP
is y x 3 1x 3. Repeat if the solution is
2.4
EXACT EQUATIONS
67
sometimes possible to find an integrating factor ␮(x, y) so that after multiplying, the
left-hand side of
␮(x, y)M(x, y) dx ␮(x, y)N(x, y) dy 0
(8)
is an exact differential. In an attempt to find ␮, we turn to the criterion (4) for exactness. Equation (8) is exact if and only if (␮M)y (␮N)x , where the subscripts
denote partial derivatives. By the Product Rule of differentiation the last equation is
the same as ␮ My ␮ y M ␮ Nx ␮ x N or
␮ x N ␮ y M (My Nx)␮.
(9)
Although M, N, My , and Nx are known functions of x and y, the difficulty here in
determining the unknown ␮(x, y) from (9) is that we must solve a partial differential
equation. Since we are not prepared to do that, we make a simplifying assumption.
Suppose ␮ is a function of one variable; for example, say that ␮ depends only on x. In
this case, ␮ x d␮dx and ␮ y 0, so (9) can be written as
d My Nx
.
dx
N
(10)
We are still at an impasse if the quotient (My Nx )N depends on both x and y.
However, if after all obvious algebraic simplifications are made, the quotient
(My Nx )N turns out to depend solely on the variable x, then (10) is a first-order
ordinary differential equation. We can finally determine ␮ because (10) is separable as well as linear. It follows from either Section 2.2 or Section 2.3 that
␮(x) e ((MyNx)/N )dx. In like manner, it follows from (9) that if ␮ depends only on
the variable y, then
d Nx My
.
dy
M
(11)
In this case, if (N x My)M is a function of y only, then we can solve (11) for ␮.
We summarize the results for the differential equation
M(x, y) dx N(x, y) dy 0.
(12)
• If (My Nx)N is a function of x alone, then an integrating factor for (12) is
(x) e
MyNx
dx
N
.
(13)
• If (Nx My)M is a function of y alone, then an integrating factor for (12) is
(y) e
NxMy
dy
M
.
(14)
EXAMPLE 4 A Nonexact DE Made Exact
The nonlinear first-order differential equation
xy dx (2x 2 3y 2 20) dy 0
is not exact. With the identifications M xy, N 2x 2 3y 2 20, we find the partial
derivatives My x and Nx 4x. The first quotient from (13) gets us nowhere, since
x 4x
3x
My Nx
2
2
2
N
2x 3y 20 2x 3y 2 20
depends on x and y. However, (14) yields a quotient that depends only on y:
Nx My
M
4x x
xy
3x
3
.
xy y
68
●
CHAPTER 2
FIRST-ORDER DIFFERENTIAL EQUATIONS
The integrating factor is then e 3dy/y e 3lny e lny y 3. After we multiply the given
DE by ␮(y) y 3, the resulting equation is
3
xy 4 dx (2x 2 y 3 3y 5 20y 3) dy 0.
You should verify that the last equation is now exact as well as show, using the
method of this section, that a family of solutions is 12 x 2 y 4 12 y 6 5y 4 c.
REMARKS
(i) When testing an equation for exactness, make sure it is of the precise
form M(x, y) dx N(x, y) dy 0. Sometimes a differential equation
is written G(x, y) dx H(x, y) dy. In this case, first rewrite it as
G(x, y) dx H(x, y) dy 0 and then identify M(x, y) G(x, y) and
N(x, y) H(x, y) before using (4).
(ii) In some texts on differential equations the study of exact equations
precedes that of linear DEs. Then the method for finding integrating factors
just discussed can be used to derive an integrating factor for
y P(x)y f (x). By rewriting the last equation in the differential form
(P(x)y f (x)) dx dy 0, we see that
M y Nx
P(x).
N
From (13) we arrive at the already familiar integrating factor e P(x)dx, used in
Section 2.3.
EXERCISES 2.4
Exatas
In Problems 1–20 determine whether the given differential
equation is exact. If it is exact, solve it.
1. (2x 1) dx (3y 7) dy 0
Answers to selected odd-numbered problems begin on page ANS-2.
12. (3x 2 y e y ) dx (x 3 xe y 2y) dy 0
13. x
2. (2x y) dx (x 6y) dy 0
3. (5x 4y) dx (4x 8y 3) dy 0
14.
1 3y x dydx y 3x 1
15.
x y
4. (sin y y sin x) dx (cos x x cos y y) dy 0
5. (2xy 2 3) dx (2x 2y 4) dy 0
6.
1
dy
y
2y cos 3x
2 4x3 3y sin 3x 0
x
dx x
7. (x y ) dx (x 2xy) dy 0
2
8.
2
2
1 ln x yx dx (1 ln x) dy
dy
2xe x y 6x 2
dx
2 3
1
dx
x 3y 2 0
2
1 9x dy
16. (5y 2x)y 2y 0
17. (tan x sin x sin y) dx cos x cos y dy 0
18. (2y sin x cos x y 2y 2e xy ) dx
2
(x sin2 x 4xye xy ) dy
2
9. (x y 3 y 2 sin x) dx (3xy 2 2y cos x) dy
10. (x 3 y 3) dx 3xy 2 dy 0
11. (y ln y e xy) dx 1y x ln y dy 0
19. (4t 3 y 15t 2 y) dt (t 4 3y 2 t) dy 0
20.
1t t1 t
2
2
y
t
dt ye y 2
dy 0
2
y
t y2
2.4
In Problems 21–26 solve the given initial-value problem.
21. (x y)2 dx (2xy x 2 1) dy 0,
22. (e x y) dx (2 x ye y ) dy 0,
y(1) 1
y(0) 1
23. (4y 2t 5) dt (6y 4t 1) dy 0, y(1) 2
3y y t dydt 2yt
2
24.
2
5
4
0,
●
69
(b) Show that the initial conditions y(0) 2 and
y(1) 1 determine the same implicit solution.
(c) Find explicit solutions y1(x) and y 2(x) of the differential equation in part (a) such that y1(0) 2
and y2(1) 1. Use a graphing utility to graph y1(x)
and y 2(x).
y(1) 1
25. (y 2 cos x 3x 2 y 2x) dx
(2y sin x x 3 ln y) dy 0,
26.
EXACT EQUATIONS
y(0) e
1
dy
cos x 2xy
y(y sin x), y(0) 1
2
1y
dx
In Problems 27 and 28 find the value of k so that the given
differential equation is exact.
27. (y 3 kxy 4 2x) dx (3xy 2 20x 2 y 3) dy 0
28. (6xy 3 cos y) dx (2kx 2y 2 x sin y) dy 0
In Problems 29 and 30 verify that the given differential equation is not exact. Multiply the given differential equation
by the indicated integrating factor ␮(x, y) and verify that the
new equation is exact. Solve.
29. (xy sin x 2y cos x) dx 2x cos x dy 0;
␮(x, y) xy
Discussion Problems
40. Consider the concept of an integrating factor used in
Problems 29–38. Are the two equations M dx N dy 0
and ␮M dx ␮N dy 0 necessarily equivalent in the
sense that a solution of one is also a solution of the other?
Discuss.
41. Reread Example 3 and then discuss why we can conclude that the interval of definition of the explicit
solution of the IVP (the blue curve in Figure 2.4.1) is
(1, 1).
42. Discuss how the functions M(x, y) and N(x, y) can be
found so that each differential equation is exact. Carry
out your ideas.
(a) M(x, y) dx xe x y 2xy 1
dy 0
x
x
(b) x1/2 y1/2 2
dx N(x, y) dy 0
30. (x 2 2xy y 2) dx (y 2 2xy x 2) dy 0;
x y
2
␮(x, y) (x y)
(vide eqs. (13) e (14) no texto acima.) 43. Differential equations are sometimes solved by
having a clever idea. Here is a little exercise in
In Problems 31 – 36 solve the given differential equation by
cleverness: Although the differential equation
finding, as in Example 4, an appropriate integrating factor.
(x 1x2 y2) dx y dy 0 is not exact, show how
2
31. (2y 3x) dx 2xy dy 0
the rearrangement (x dx y dy) 1x2 y2 dx and
the observation 12 d(x 2 y 2) x dx y dy can lead to
32. y(x y 1) dx (x 2y) dy 0
a solution.
33. 6xy dx (4y 9x 2) dy 0
44. True or False: Every separable first-order equation
dydx g(x)h(y) is exact.
2
34. cos x dx 1 sin x dy 0
y
35. (10 6y e3x ) dx 2 dy 0
36. (y 2 xy 3) dx (5y 2 xy y 3 sin y) dy 0
In Problems 37 and 38 solve the given initial-value problem
by finding, as in Example 4, an appropriate integrating factor.
37. x dx (x 2 y 4y) dy 0,
y(4) 0
38. (x 2 y 2 5) dx (y xy) dy,
y(0) 1
39. (a) Show that a one-parameter family of solutions of
the equation
(4xy 3x 2) dx (2y 2x 2) dy 0
is x 3 2x 2 y y 2 c.
Mathematical Model
45. Falling Chain A portion of a uniform chain of length
8 ft is loosely coiled around a peg at the edge of a high
horizontal platform, and the remaining portion of the
chain hangs at rest over the edge of the platform. See
Figure 2.4.2. Suppose that the length of the overhanging chain is 3 ft, that the chain weighs 2 lb/ft, and that
the positive direction is downward. Starting at t 0
seconds, the weight of the overhanging portion causes
the chain on the table to uncoil smoothly and to fall to
the floor. If x(t) denotes the length of the chain overhanging the table at time t 0, then v dxdt is its
velocity. When all resistive forces are ignored, it can
be shown that a mathematical model relating v to x is
74
●
CHAPTER 2
FIRST-ORDER DIFFERENTIAL EQUATIONS
EXERCISES 2.5
Homogêneas
Answers to selected odd-numbered problems begin on page ANS-2.
Each DE in Problems 1 – 14 is homogeneous.
Each DE in Problems 23 – 30 is of the form given in (5).
In Problems 1–10 solve the given differential equation by
using an appropriate substitution.
In Problems 23 – 28 solve the given differential equation by
using an appropriate substitution.
1. (x y) dx x dy 0
2. (x y) dx x dy 0
3. x dx (y 2x) dy 0
4. y dx 2(x y) dy
5. (y 2 yx) dx x 2 dy 0
23.
dy
(x y 1) 2
dx
24.
dy 1 x y
dx
xy
25.
dy
tan2 (x y)
dx
26.
dy
sin(x y)
dx
dy
2 1y 2x 3
dx
28.
dy
1 eyx5
dx
6. (y 2 yx) dx x 2 dy 0
7.
dy y x
dx y x
27.
dy x 3y
dx 3x y
In Problems 29 and 30 solve the given initial-value problem.
8.
(
29.
dy
cos(x y), y(0) > 4
dx
30.
dy
3x 2y
,
dx 3x 2y 2
)
9. y dx x 1xy dy 0
dy
10. x
y 1x2 y2,
dx
x0
In Problems 11 – 14 solve the given initial-value problem.
11. xy2
dy
y3 x3,
dx
12. (x 2 2y 2)
dx
xy, y(1) 1
dy
dy
y
.
F
dx
x
y(1) 0
14. y dx x(ln x ln y 1) dy 0,
y(1) e
Each DE in Problems 15 – 22 is a Bernoulli equation.
In Problems 15 – 20 solve the given differential equation by
using an appropriate substitution.
15. x
17.
dy
1
y 2
dx
y
dy
y(xy 3 1)
dx
19. t2
dy
y2 ty
dt
16.
dy
y ex y2
dx
18. x
dy
(1 x)y xy2
dx
20. 3(1 t2)
dy
2ty( y3 1)
dt
In Problems 21 and 22 solve the given initial-value problem.
21. x2
Discussion Problems
31. Explain why it is always possible to express any homogeneous differential equation M(x, y) dx N(x, y) dy 0 in
the form
y(1) 2
13. (x ye y/x ) dx xe y/x dy 0,
y(1) 1
You might start by proving that
M(x, y) xa M(1, y>x)
and
N(x, y) x aN(1, y>x).
32. Put the homogeneous differential equation
(5x 2 2y 2) dx xy dy 0
into the form given in Problem 31.
33. (a) Determine two singular solutions of the DE in
Problem 10.
(b) If the initial condition y(5) 0 is as prescribed in
Problem 10, then what is the largest interval I over
which the solution is defined? Use a graphing utility to graph the solution curve for the IVP.
34. In Example 3 the solution y(x) becomes unbounded as
x : . Nevertheless, y(x) is asymptotic to a curve as
x : and to a different curve as x : . What are the
equations of these curves?
dy
2xy 3y4,
dx
y(1) 12
35. The differential equation dydx P(x) Q(x)y R(x)y2
is known as Riccati’s equation.
dy
y3/2 1,
dx
y(0) 4
(a) A Riccati equation can be solved by a succession
of two substitutions provided that we know a
22. y1/2
ANSWERS FOR SELECTED
ODD-NUMBERED PROBLEMS
1.
5.
9.
15.
17.
19.
27.
33.
linear, second order
3. linear, fourth order
nonlinear, second order 7. linear, third order
linear in x but nonlinear in y
domain of function is [2, ); largest interval of
definition for solution is (2, )
domain of function is the set of real numbers except
x 2 and x 2; largest intervals of definition for
solution are (
, 2), (2, 2), or (2, )
et 1
defined on (
, ln 2) or on (ln 2, )
X t
e 2
m 2
29. m 2, m 3
31. m 0, m 1
y2
35. no constant solutions
EXERCISES 1.2 (PAGE 17)
1.
3.
5.
7.
y 1(1 4ex )
y 1(x 2 1); (1, )
y 1(x 2 1); (
, )
x cos t 8 sin t
9. x 13
4
15. y 0, y x 3
y 5ex1
half-planes defined by either y 0 or y 0
half-planes defined by either x 0 or x 0
the regions defined by y 2, y 2, or 2 y 2
any region not containing (0, 0)
yes
no
(a) y cx
(b) any rectangular region not touching the y-axis
(c) No, the function is not differentiable at x 0.
31. (b) y 1(1 x) on (
, 1);
y 1(x 1) on (1, );
(c) y 0 on (
, )
EXERCISES 1.3 (PAGE 27)
3.
7.
9.
11.
dP
dP
kP r;
kP r
dt
dt
dP
k1 P k2 P2
dt
dx
kx(1000 x)
dt
1
dA
A 0; A(0) 50
dt
100
dh
dA
7
c
13.
A6
1h
dt
600 t
dt
450
dv
mg kv2
dt
d 2r gR 2
21. 2 2 0
dt
r
dx
kx r, k 0
25.
dt
17. m
CHAPTER 1 IN REVIEW (PAGE 32)
1.
5.
9.
13.
15.
17.
19.
cos t 14 sin t 11. y 32 ex 12 ex
13.
17.
19.
21.
23.
25.
27.
29.
1.
di
Ri E(t)
dt
d 2x
19. m 2 kx
dt
dA
k(M A), k 0
23.
dt
dy x 1x2 y2
27.
dx
y
15. L
21.
dy
10y
3. y k 2 y 0
dx
y 2 y y 0
7. (a), (d)
(b)
11. (b)
y c 1 and y c 2e x, c 1 and c 2 constants
y x 2 y 2
(a) The domain is the set of all real numbers.
(b) either (
, 0) or (0, )
For x 0 1 the interval is (
, 0), and for x 0 2 the
interval is (0, ).
x2, x 0
(c) y 2
23. (
, )
x,
x0
25. (0, )
27. y 12 e3x 12 ex 2x
29. y 32 e3x3 92 ex1 2x.
31. y 0 3, y 1 0
33.
dP
k(P 200 10t)
dt
EXERCISES 2.1 (PAGE 41)
21. 0 is asymptotically stable (attractor); 3 is unstable
(repeller).
23. 2 is semi-stable.
25. 2 is unstable (repeller); 0 is semi-stable; 2 is
asymptotically stable (attractor).
27. 1 is asymptotically stable (attractor); 0 is unstable
(repeller).
39. 0 P0 hk
41. 1mg>k
EXERCISES 2.2 (PAGE 50)
1. y 15 cos 5x c
3. y 13 e3x c
5. y cx 4
7. 3e2y 2e 3x c
9.
1 3
3 x ln
x 19 x3 12 y2 2y ln y c
ANS-1
ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS • CHAPTER 2
EXERCISES 1.1 (PAGE 10)
ANS-2
ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS
●
EXERCISES 2.4 (PAGE 68)
11. 4 cos y 2x sin 2x c
13. (e x 1) 2 2(e y 1) 1 c
t
ce
17. P 1 cet
15. S ce kr
19. ( y 3) 5 e x c(x 4) 5 e y
(
23. x tan 4t 34 13
2
27. y 12 x 21. y sin
)
25. y ( 12 x2 c)
e(11/x)
x
x -t dt
29. y e4 e
2
11 x2
3 e4 x1
3 e4 x1
33. y 1 and y 1 are singular solutions of Problem 21;
y 0 of Problem 22
35. y 1
ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS • CHAPTER 2
31. (a) y 2, y 2, y 2
37. y 1 101 tan
(101 x)
(
41. (a) y 1x2 x 1
(c) , 12 12 15
49. y(x) (4hL2 )x 2 a
)
1. x2 x 32 y2 7y c
3.
5. x 2 y 2 3x 4y c
7. not exact
xy y cos x x c
not exact
xy 2xe x 2e x 2x 3 c
x 3 y 3 tan1 3x c
ln cos x cos x sin y c
t 4 y 5t 3 ty y 3 c
21.
1 3
3x
23.
25.
27.
31.
35.
4ty t 2 5t 3y 2 y 8
y 2 sin x x 3 y x 2 y ln y y 0
k 10
29. x 2 y 2 cos x c
2 2
3
33. 3x 2 y 3 y 4 c
x y x c
2ye3x 103 e3x x c
x
x3
, (
, ); ce
1
37. ey (x2 4) 20
2
is transient
is transient
1
7. y x ln x cx , (0, ); solution is transient
9. y cx x cos x, (0, )
11. y 17 x3 15 x cx4, (0, ); cx4 is transient
13. y 15.
17.
19.
21.
23.
25.
1 2 x
2x e
cx2 ex, (0, ); cx2ex is transient
x 2y 6 cy 4, (0, )
y sin x c cos x, (p2, 2)
(x 1)e xy x 2 c, (1, ); solution is transient
(sec u tan )r u cos u c, ( 2, p2)
y e3x cx 1e3x, (0, ); solution is transient
y x 1e x (2 e)x 1, (0, )
E
E Rt /L
27. i i0 e
, (
, )
R
R
29. (x 1)y x ln x x 21, (0, )
31. y 33. y (
35. y 2x4x ln x1 (14 e 4e,
1
2 x
),
2 (1 e
1 6
2 x
,
2 (e 1)e
32 ex ,
1
2
37. y ex
2
1
2e
3
2
)e
0x3
x3
0x1
, x1
x 2
2x
2
2
2
1
0x1
)x2, x 1
12 1 ex (erf(x) erf(1))
47. E(t) E 0 e(t4)/RC
2
2
x2 y xy2 y 43
45. (a) v(x) 8
3. y e ce , (
, ); ce
5. y ce
1
2
2
y2 (x) x2 1x4 x3 4
x
x 3
3
39. (c) y1 (x) x2 1x4 x3 4
1. y ce 5x, (
, )
3x
4xy 2 y4 c
9.
11.
13.
15.
17.
19.
EXERCISES 2.3 (PAGE 60)
1
4
1
3
5 2
2x
x
9
B3 x2
(b) 12.7 ft/s
EXERCISES 2.5 (PAGE 74)
1. y x lnx cx
3. (x y)ln x y y c(x y)
5. x y lnx cy
7.
9.
13.
17.
ln(x 2 y 2 ) 2 tan1( yx) c
4x y(lny c) 2
11. y 3 3x 3 lnx 8x 3
y/x
lnx e 1
15. y 3 1 cx3
1
3
3x
y x 3 ce
19. e t/y ct
21. y3 95 x1 495 x6
y x 1 tan(x c)
2y 2 x sin 2(x y) c
4( y 2x 3) (x c) 2
cot(x y) csc(x y) x 12 1
2
35. (b) y 14 x cx3 1
x
23.
25.
27.
29.
(
)
EXERCISES 2.6 (PAGE 79)
1.
3.
5.
7.
9.
13.
y 2 2.9800, y 4 3.1151
y10 2.5937, y 20 2.6533; y e x
y5 0.4198, y10 0.4124
y5 0.5639, y10 0.5565
y5 1.2194, y10 1.2696
Euler: y10 3.8191, y 20 5.9363
RK4: y10 42.9931, y 20 84.0132