130 CHAPTER 4 HIGHERORDER DIFFERENTIAL EQUATIONS

in Theorem 4.1.1, the output or response y(t) is uniquely determined by the

input and the state of the system prescribed at a time t0 —that is, by the initial
conditions y(tt0), y9(tt0), . . . , y (n−1)(t 0).
For a dynamical system to be a linear system, it is necessary that the
superposition principle (Theorem 4.1.7) holds in the system; that is, the
response of the system to a superposition of inputs is a superposition of out-
puts. We have already examined some simple linear systems in Section 3.1
(linear first-order equations); in Section 5.1 we examine linear systems in
which the mathematical models are second-order differential equations.

EXERCISES 4.1 Answers to selected odd-numbered problems begin on page ANS-4.

4.1.1 Initial-Value and Boundary-Value Problems 10. y0 + (tan x)y = ex, y(0) = 1, y9(0) = 0

In Problems 1 – 4 the given family of functions is the general solution 11. (a) Use the family in Problem 1 to find a solution of y0 − y = 0
of the differential equation on the indicated interval. Find a member that satisfies the boundary conditions y(0) = 0, y(1) = 1.
of the family that is a solution of the initial-value problem. (b) The DE in part (a) has the alternative general solution
1. y = c1ex + c2e−x, (−`, `); y = c3 cosh x + c4 sinh x on (−`, `). Use this family to
find a solution that satisfies the boundary conditions in
y0 − y = 0, y(0) = 0, y9(0) = 1
part (a).
2. y = c1e4x + c2e−x, (−`, `);
(c) Show that the solutions in parts (a) and (b) are equivalent
y0 − 3y9 − 4y
4 = 0, y(0) = 1, y9(0) = 2
12. Use the family in Problem 5 to find a solution of
3. y = c1x + c2x ln x, (0, `); xy 0 − y9 = 0 that satisfies the boundary conditions
x 2y0 − xy9 + y = 0, y(1) = 3, y9(1) = −1 y(0) = 1, y9(1) = 6.

4. y = c1 + c2 cos x + c3 sin x, (−`, `);

In Problems 13 and 14 the given two-parameter family is a solution
y- + y9 = 0, y() = 0, y9() = 2, y0() = −1 of the indicated differential equation on the interval (−`, `). Deter-
mine whether a member of the family can be found that satisfies the
5. Given that y = c1 + c2x2 is a two-parameter family of solutions boundary conditions.
of xy0 − y9 = 0 on the interval (−`, `), show that constants c1
and c2 cannot be found so that a member of the family satisfies 13. y = c1ex cos x + c2ex sin x; y0 − 2y9 + 2y = 0
the initial conditions y(0) = 0, y9(0) = 1. Explain why this does
(a) y(0) = 1, y9() = 0
not violate Theorem 4.1.1.
(b) y(0) = 1, y() = −1
6. Find two members of the family of solutions in Problem 5 that
satisfy the initial conditions y(0) = 0, y9(0) = 0. (c) y(0) = 1, y(y2) 5 1
7. Given that x(t) = c1 cos t + c2 sin t is the general solution (d) y(0) = 0, y() = 0.
of x0 + 2x = 0 on the interval (−`, `), show that a solution
satisfying the initial conditions x(0) = x0, x9(0) = x1 is 14. y = c1x + c2x + 3; x2y0 − 5xy9 + 8y = 24
2 4

given by (a) y(−1) = 0, y(1) = 4

x1
x(t) 5 x0 cos t 1 sin t. (b) y(0) = 1, y(1) = 2

(c) y(0) = 3, y(1) = 0
8. Use the general solution of x0 + 2x = 0 given in
Problem 7 to show that a solution satisfying the initial (d) y(1) = 3, y(2) = 15
conditions x(tt0) = x0, x9(tt0) = x1 is the solution given in
Problem 7 shifted by an amount t0:
4.1.2 Homogeneous Equations
x1
x(t) 5 x0 cos (t 2 t0) 1 sin (t 2 t0). In Problems 15 – 22 determine whether the given set of functions is

linearly independent on the interval (−`, `).
In Problems 9 and 10 find an interval centered about x = 0 for which
the given initial-value problem has a unique solution. 15. f1(x) = x, f2(x) = x 2, f3(x) = 4x − 3x 2

9. (x − 2)y0 + 3y = x, y(0) = 0, y9(0) = 1 16. f1(x) = 0, f2(x) = x, f3(x) = e x

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 4.1 PRELIMINARY THEORYLINEAR EQUATIONS 131

17. f1(x) = 5, f2(x) = cos2 x, f3(x) = sin2 x (b) By inspection find a particular solution of

18. f1(x) = cos 2x, f2(x) = 1, f3(x) = cos2 x y0 + 2y = −4x.

19. f1(x) = x, f2(x) = x − 1, f3(x) = x + 3 (c) Find a particular solution of y0 + 2y = −4x + 10.
(d) Find a particular solution of y0 + 2y = 8x + 5.
20. f1(x) = 2 + x, f2(x) = 2 + u xu
21. f1(x) = 1 + x, f2(x) = x, f3(x) = x 2
Discussion Problems
22. f1(x) = ex, f2(x) = e−x, f3(x) = sinh x
37. Let n = 1, 2, 3, . . . . Discuss how the observations Dnxn−1 = 0
In Problems 23–30 verify that the given functions form a fundamen- and Dnxn = n! can be used to find the general solutions of the
tal set of solutions of the differential equation on the indicated inter- given differential equations.
val. Form the general solution.
(a) y0 = 0 (b) y- = 0 (c) y(4) = 0
−3x 4x
23. y0 − y9 − 12y = 0; e , e , (−`, `)
(d) y0 = 2 (e) y- = 6 (f) y(4) = 24
24. y0 − 4y = 0; cosh 22x, sinh 22x, (−`, `)
38. Suppose that y1 = ex and y2 = e−xx are two solutions of
x x
25. y0 − 2y9 + 5y = 0; e cos 22xx, e sin 22xx, (−`, `) a homogeneous linear differential equation. Explain why
y3 = cosh x and y4 = sinh x are also solutions of the
26. 4y0 − 4y9 + y = 0; ex/2
x
, xex/2
x
, (−`, `) equation.
27. x2y0 − 6xy9 + 12y = 0; x3, x4, (0, `) 39. (a) Verify that y1 = x3 and y2 = u x u3 are linearly independent
2
28. x y0 + xy9 + y = 0; cos(ln x), sin(ln x), (0, `) solutions of the differential equation x2y0 − 4xy9 + 6y = 0
on the interval (−`, `).
29. x3y- + 6x2y0 + 4xy9 − 4y = 0; x, x−2, x−2 ln x, (0, `)
(b) For the functions y1 and y2 in part (a), show that
30. y(4) + y0 = 0; 1, x, cos x, sin x, (−`, `) W y1, y2) 5 0 for every real number x. Does this result
W(
violate Theorem 4.1.3? Explain.
4.1.3 Nonhomogeneous Equations
(c) Verify that Y1 = x3 and Y2 = x2 are also linearly
In Problems 31 – 34 verify that the given two-parameter family of independent solutions of the differential equation in part (a)
functions is the general solution of the nonhomogeneous differential on the interval (−`, `).
equation on the indicated interval.
(d) Besides the functions y1, y2, Y1, and Y2 in parts (a) and (c),
31. y0 − 7y9 + 10y = 24ex; find a solution of the differential equation that satisfies
y = c1e2x
2
+ c2e5x + 6ex, (−`, `) y(0) 5 0, y9(0) 5 0.

32. y0 + y = sec x; (e) By the superposition principle, Theorem 4.1.2, both linear
combinations y = c1y1 + c2y2 and Y = c1Y1 + c2Y2 are
y = c1 cos x + c2 sin x + x sin x + (cos x) ln(cos x), (−y2, y2)
solutions of the differential equation. Discuss whether
33. y0 − 4y9 + 4y = 2e 2x + 4x − 12; one, both, or neither of the linear combinations is a
y = c1e 2x + c2 xe 2x + x 2e 2x + x − 2, (−`, `) general solution of the differential equation on the
interval (−`, `).
2 2y0 + 5xy9 + y = x2 − x;
34. 2x
1 2
y 5 c1x21/2 1 c2 x21 1 15 x 2 16x, (0, `) 40. Is the set of functions f1(x) = ex+2, f2(x) = ex−3 linearly
dependent or linearly independent on (−`, `)? Discuss.
35. (a) Verify that yp1 5 3e2x and yp2 5 x2 1 3x are, respectively,
particular solutions of 41. Suppose y1, y2, . . . , yk are k linearly independent solutions
on (−`, `) of a homogeneous linear nth-order differential
y0 2 6y9 1 5y 5 29e2x equation with constant coefficients. By Theorem 4.1.2 it
follows that yk+1 = 0 is also a solution of the differential
and y0 2 6y9 1 5y 5 5x2 1 3x 2 16. equation. Is the set of solutions y1, y2, . . . , yk, yk+1
linearly dependent or linearly independent on (−`, `)?
(b) Use part (a) to find particular solutions of
Discuss.
y0 2 6y9 1 5y 5 5x2 1 3x 2 16 2 9e2x
42. Suppose that y1, y2, . . . , yk are k nontrivial solutions of a
and y0 2 6y9 1 5y 5 210x 2 2 6x 1 32 1 e2x. homogeneous linear nth-order differential equation with
constant coefficients and that k = n + 1. Is the set of solutions
36. (a) By inspection find a particular solution of
y1, y2, . . . , yk linearly dependent or linearly independent on
y0 + 2y = 10. (−`, `)? Discuss.

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