Exercise 4.2
2. Solution: y2 = e–x.
11. Solution: y2 = –1. Hint: integration by change of variable u = ln x.
24. Solution: 𝑦 = 𝑐 𝑥 ∫ 𝑑𝑥 = 𝑐 −1 + 𝑥 𝑙𝑛 𝑥 +
!𝑥 +
( !)𝑥 +
( !)𝑥 + ⋯ , for x
(0, ).
Exercise 4.3
12. Solution: y = e–x/2[c1cos(x/2) + c2 sin(x/2)].
22. Solution: y = c1 e2x + c2 xe2x + c3 x2e2x. 30. Solution: y 3cos sin.
59. Solution: c[y – (11/2)y + 7y + 5y ] = 0, for any c R\{0}.
Exercise 4.4
2. Solution: y = c1 cos (3/2)x + c2 sin(3/2)x + 5/3.
34. Solution: F t
F t
x
cos 2 2 cos
0 2
2 0
.
38. Solution: y = c2 ex sin x + x.
