general solution using laplace, convolution

Find the general solution to

y'' + 9y =g(t), y(0)=0, y'(0)=1 using Laplace Transform.

Express your result in the form of the convolution of g*e where e is the unit impulse response for the system modeled by the above equation.



This is what I did:

(s^2)Y(s)-sy(0)-y'(0) +9[sY(s)-y(0)]

(s^2)Y(s)-s(0)-1 +9[sY(s)-(0)]= g(t)

Y(s)[s^2+9]-1=g(t), =0

Y(s)= s/(s^2+9)= 1/(s^2+3^2)

= cos(3t)



What did I do wrong?

Review the laplace transform tables. 9y -> 9Y. 1/3*3/(s^2 + 3^2) -> 1/3*sin(3t), etc. g(t) disappeared somehow...

oh....

thanks!