Solving y''-6y+9y=2 with Laplace Transforms

[f00lishroy]

Homework Statement

y''-6y+9y=2 y(0)=y'(0)=0

*Note* Professor will NOT allow use of partial fractions, so please don't use it.

Homework Equations

Laplace transform table
Y=[y'(0)+sy(0)+ay(0)+R]/[(s^2)+as+b]

The Attempt at a Solution

Y=L(2)/(s-3)^2
L(2)=2/s
Y=(2/s)[1/(s-3)^2]
Y=2*1/[s^3-6s^2+9s]
From here I cannot figure out how to continue without using partial fractions since I can't get the roots, from which I would be able to invert and use the Laplace Transform table.

 

[lanedance]

so you have
$$Y(s) = F(s)G(s)$$
with
$$F(s)=\frac{2}{s}$$
$$G(s)=\frac{1}{(s-3)^2}$$

how about consider f(t) and g(t) and using a convolution, if you don't know how to do this have a look at 1.7 in
http://www.vibrationdata.com/math/Laplace_Transforms.pdf

 

[vela]

You could also invert Y(s) using the Bromwich integral
$$y(t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} Y(s)e^{st}\,ds$$.