Is this going in the right direction with this problem?

$\displaystyle y''-4y'+13y=e^{3t}; y(0)=0; y'(0)=1$

$\displaystyle s^2Y(s)-1-4sY(s)+13Y(s)=\frac{1}{s-3}$

$\displaystyle Y(s)[s^2-4s+13]=\frac{1}{s-3}+1$

$\displaystyle Y(s)=\frac{1}{(s-3)[(s-2)^2+11]}+\frac{1}{(s-2)^2+11}$

Before I continue down a futile path, should this work or is there a more efficient way?

I don't need an answer just a yes this way is fine or no try this, thanks.