Differential equation with Laplace transform

[springo]

Homework Statement

t y'' + (3 t-1) y' + 3 y = 6 e^{-3t}
y(0) = 1
y(5) = 2

Homework Equations

The Attempt at a Solution

I tried applying the Laplace transform to the equation but I was having a little trouble...
L [t·y''] = -dL[y'']/ds = s²·Y' + 2·s·Y - y(0) = s²·Y' + 2·s·Y - 1
L [t·y'] = -dL[y']/ds = s·Y' + Y
Therefore:
s^2 Y' + 2 s Y - 1 + 3 s Y' +3 Y - s Y + 1+ 3 Y = \frac{6}{s+3}
(s^2+3s)Y'+(s+6)Y=\frac{6}{s+3}

And then solve the Y(s) and then use the inverse transform.
But it doesn't yield the correct result. I did the rest on the computer so I guess it's when I apply the transform where I make a mistake.

Thanks for your help.

 

[vela]

I think you dropped the negative sign when calculating L[t y''] and L[t y'].