Find the particular solutions using Laplace transforms If ne

Find the particular solutions using Laplace transforms. If necessary, use partial fraction expansion. Show your work.

y\'\'+4y=0, y(0)=1, y(pi/4)=1

Solution

y”+4y=0    y(0)=1   y(pi/4)=0

Taking Laplace transform on both sides

L((y”(0) + 4y(0)) = L(0)

Ly”(0) + 4 L(y(0)) = L (0)

Use,

                L(y”) = S²L(y) - Sy(0) - y\'(0)

S²y(S) - Sy(0) - y(0) +4y(S) = 0

Put y(0) = 1

S² y(S) - 1 - 1 + 4y(S) = 0

S² y(S) + 4y(S) - 2 = 0

(S² + 4) y(S) = 2

y(S) = 2 / (S² + 4)

= 2((1/ (S² + 2²))

y(S) = 2 Sin 2t