Transform the differential equation $$2y'''+12y''-72y'-432y=8e^{-2t}$$ $$y(0)=0$$ $$y^\prime(0)=0$$ $$y^{\prime\prime}(0) = 1$$ into an algebraic equation by taking the Laplace transform of each side. Use $Y$ for the Laplace transform of $y$, (not $Y(s)$).

Therefore

$\displaystyle Y =$

    $\displaystyle =$ $\frac{1}{s-6}+$$\frac{1}{s+6}+$$\frac{1}{s+2}$

Taking the inverse Laplace transform we get
$y =$