# Math Help - Laplace Transforms

1. ## Laplace Transforms

use Laplace transforms to solve the inital value problem

y'' + 5y' + 4y = 2e^-3t y(0) = 0 & y'(0) = -1

thanks

2. $y''+5y'+4y=2e^{-3t}, \;\ y(0)=0, \;\ y'(0)=-1$

$p^{2}Y-py(0)-y'(0)+5(pY-y(0))+4Y=\frac{2}{p+3}$

$p^{2}Y+1+5pY+4Y=\frac{2}{p+3}$

$Y(p^{2}+5p+4)=\frac{2}{p+3}-1$

$Y=\frac{1}{p+4}-\frac{1}{p+3}$

Now use the inverse LaPlace:

$y=e^{-4t}-e^{-3t}$