Page 261 - Schaum's Outline of Differential Equations

244 SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS [CHAP. 24

Solving for Y(s), we find

Taking the inverse Laplace transform, and using the result of Problem 22.17, we obtain

24.4. Solve /' + 4y = 0; y(0) = 2, /(O) = 2.
Taking Laplace transforms, we have ££{j"} + 4!£{y} = ££{0}. Then, using Eq. (24.5) with c 0 = 2 and c 1 = 2,
we obtain

or

Finally, taking the inverse Laplace transform, we obtain

24.5. Solve /'- 3/ + 4y = 0; y(0) = 1, /(O) = 5.
Taking Laplace transforms, we obtain ££{/'} -3£6{/} + 4£6{;y} =£6{0}. Then, using foort Eqs. (24.4) and
(24.5) with c 0 = 1 and Cj = 5, we have

or

Finally, taking the inverse Laplace transform and using the result of Problem 22.10, we obtain

2
24.6. Solve /'- / - 2y = 4x ; y(0) = 1, /(O) = 4.
i
Taking Laplace transforms, we have ££{/'}-££{/} -2%{y} =4%{x }. Then, using both Eqs. (24.4) and
(24.5) with c 0 = 1 and Cj = 4, we obtain

or, upon solving for y(j),

Finally, taking the inverse Laplace transform and using the results of Problems 22.15 and 22.16, we obtain

(See Problem 13.1.)