Taking Laplace transform of both sides gives:

`(sY-y(0))+Y=3/(s^2+9)`

`sY+Y=3/(s^2+9)` (since `y(0) = 0`)

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

Solving for Y and finding the partial fraction decomposition gives:

`Y=3/((s+1)(s^2+9))` `=A/(s+1)+(Bs+C)/(s^2+9)`

`3=A(s^2+9)+(s+1)(Bs+C)`

Substituting convenient values of `s` gives us:

`s=-1` gives `3=10A`, which gives `A=3/10`.

`s=0` gives `3=9A+C`, which gives `C=3/10`.

`s=1` gives `3=10A+2B+2C`, which gives us `B=-3/10`.

So

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

`=3/10(1/(s+1)+(-s+1)/(s^2+9))`

`=3/10(1/(s+1)-s/(s^2+9)+1/(s^2+9))`

Finding the inverse Laplace tranform gives us the solution for y as a function of t:

`y=3/10e^(-t)-3/10cos\ 3t+1/10sin\ 3t`

Please support IntMath!