Page 262 - Schaum's Outline of Differential Equations

CHAP. 24] SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS 245

24.7. Solve /' + 4/ + 83; = sin x; y(0) = 1, /(O) = 0.
c
Taking Laplace transforms, we obtain £(y"} + 4££{/} + &£{y} = ££{sin x}. Since c 0 = 1 and Cj=0, this
becomes

Thus,

Finally, taking the inverse Laplace transform and using the results of Problems 22.9 and 22.18, we obtain

(See Problem 13.3.)

24.8. Solve /'- 2/ + y = f(x); y(0) = 0, /(O) = 0.
In this equation/I*) is unspecified. Taking Laplace transforms and designating ££{/(jc)} by F(s), we obtain

2
1
x
From Appendix A, entry 14, ££ {l/(5— I) } =xe . Thus, taking the inverse transform of Y(s) and using convolutions,
we conclude that

24.9. Solve /' + y =f(x); y(0) = 0, /(O) = 0 if /(*) =

Note that/(jc) = 2u (x — 1). Taking Laplace transforms, we obtain

or

Since

it follows from Theorem 23.4 that

24.10. Solve /" +/ = (?; y(0) = /(O) = y"(0) = 0.
Taking Laplace transforms, we obtain ££{/"} + ££{/} = £((?). Then, using Eq. (24.3) with n = 3 and Eq. (24.4),
we have