Solve each of the following DE subject to given conditions i

Solve each of the following DE subject to given conditions, if any. Use the Laplace transform to solve the initial value problem: y\" + 4y - 8 sin (2t) + 9 cos t, y(0) = 1, y\'(0) = 0.

Solution

u(x, y) = b0 y H + X n=1 bn sinh nH L 1 sinh ny L cos nx L , where b0 + X n=1 bn cos nx L is the Fourier cosine series of the function f(x) on [0, L], that is, b0 = 1 L Z L 0 f(x) dx, bn = 2 L Z L 0 f(x) cos nx L dx, n = 1, 2, . . .