Laplace Transform of IVP

Question

I have this equation

(d^2y/dt^2) + k^2y = e^(1t) , y(0 = 0, y'(0) = 0

I need to simplify Y, so far I have this:

s^2Y+k^2Y = 1/(s-1)

Y = (1) / (s-1)(s^2+k^2)

I need to get Y in simpler forms

= ( ) / (s-1) +(( ) s + ( ) / (s^2+k^2))

Answer 1

you can take 1/(s-1)(s² + k²) and break it up into A/(s-1) + (Bs + C)/(s² + k²) to get

A = 1/(1 + k²), B = -1/(1 + k²), and C = -1/(1 + k²) Let 1/(1 + k²) = p = constant

then you have Y(s) = p/(s-1) - ps/(s² + k²) - p/(s² + k²)

to find that y(t) = pe^t - pcos(kt) - (p/k)sin(kt)