Let $a,b,c \in \mathbb R$ with $a \neq 0$. Solve the IVP
$\displaystyle ay'' + by' + cy = t$
Using Laplace Transform where $\displaystyle y'(0)=0,y(0)=1$
My Attempt
$\displaystyle a\mathcal L(y'') + b \mathcal L(y')+ c\mathcal L(y) = \mathcal L t$
$\displaystyle a[s^2y(s)-sy(0)-y'(0)] + b[sy(s)-y(0)]+cy(s) = \frac{1}{s^2}$
$\displaystyle y(s)(as^2 + bs +c) -asy(0)-by(0) - ay'(0) = \frac{1}{s^2}$
Plugging initial values $y'(0)=0$ and $y(0)=1$ we have:
$\displaystyle y(s) = \dfrac{as+b}{s^2(as^2+bs+c)}$
Now for the next part where I'm stuck with obtaining partial fractions and using Inverse Laplace since the constants are $a,b,c$. Any help would be appreciated.