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

y'' + k²y = e^t;

s²Y + k²Y = 1/(s - 1); Y = 1/[(s - 1)(s² + k²) = a/(s - 1) + (bs + c)/(s² + k²) = 1//(k² + 1)[1/(s - 1) - (s + 1)/(s² + k²)]; y = L^-1(Y) = 1/(k² + 1)[e^t - cos(kt) - 1/ksin(ka)]

Y(s) = 1/(k² + 1)[1/(s - 1) - s/(s² + k²) - 1(s² + k²)]