Take the Laplace transform of the IVP

Question

This is my equation, dy/dt - y = -10cos(kt), y(0)=3

I need to find what Y is and then what y(t) is

I know the right hand side is -10 * s/(s^2+k^2)

Answer 1

sY(s) - 3 - Y(s) = - 1 0s/(s² + k²); Y(s) = 3/(s - 1) - 10s/[(s² + k²)(s - 1)] = 3/(s- 1) - 10/(k² +1)·1/(s - 1) +

10/(k² + 1)·s/(s² + k²) - 10k/(k² + 1)·1/(s² + k²)

y(t) = L^-1(Y(s)) = 3e^s - 10/(k²+ 1)e^s + 10/(k² + 1)coskt - 10(k²+ 1)sinkt