Page 96 - Schaum's Outline of Differential Equations

CHAP. 8] LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS 79

8.22. Use the results of Problem 8.18 to find the general solution of

if it is known that x + 4x + 6 is a particular solution.
We have from Problem 8.18 that the general solution to the associated homogeneous differential equation is

2
Since we are given that y p = x + 4x + 6, it follows from Theorem 8.4 that

8.23. Use the results of Problem 8.18 to find the general solution of

if it is known that ±e 3x is a particular solution.
We have from Problem 8.18 that the general solution to the associated homogeneous differential equation is

3
In addition, we are given that y p = -^e *. It follows directly from Theorem 8.4 that

3
3
$.24. Determine whether the set {.x , be !} is linearly dependent on [—1, 1].
Consider the equation
3
3
3
Recall that U ! = x 3 if x > 0 and bt ! = -x if x < 0. Thus, when x>0,(l) becomes
whereas when x < 0, (1) becomes

Solving (2) and (3) simultaneously for c l and c 2, we find that the only solution is c l = c 2 = 0. The given set is, therefore,
linearly independent.

8.25. Find
We have

Then, for x> 0,

For x < 0,

For x = 0,

Thus,