Laplace Transform IVP, trouble getting inverse transform.

Question

I am given the IVP $y^{(4)}-4y'''+6y''-4y'+y=0,\ y(0)=0,\ y'(0)=1,\ y''(0)=0,\ y'''(0)=1.$ I have taken the Laplace transform of all the terms, and have ended up with $$F(s)=\frac{(s-2)^2}{(s-1)^4}+\frac{3}{(s-1)^4}.$$ Do I have to use partial fractions to decompose these further, or is there a simpler way to simplify these expressions?

Answer 1

The answer is

$$ \frac{2{{t}^{3}}\,{{ {}e}^{t}}}{3}-{{t}^{2}}\,{{ {}e}^{t}}+t\,{{ {}e}^{t}} $$ you can use any computer algebra system.

Answer 2

If you can write $F(s)$ as the derivative or integral of some other function $G(s)$ then you can use #30 or #31 from: http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf

Otherwise you need to use partial fraction decomposition.