TITLE: Solving a differential equation with the Dirac-Delta function without Laplace transformations
QUESTION [6 upvotes]: So I'm trying to solve the following differential equation:
$y''+3y'+2y=\delta(t-1)$, $y(0)=0$, $y'(0)=0$. (where $\delta$ is the Dirac's delta function)
Everything I've read in my textbook/online has solved these types of equations by taking the Laplace transformation, but our class hasn't covered Laplace transformations yet...anyone have any idea what I should do?

REPLY [7 votes]: The physicist's answer, not worrying about convergence, uniqueness, etc.  It sounds like you have no problem away from $t=1$.  Up to $1^-,\ \ y(t)=0$.  Above $1$, you can solve it with $y$ being a sum of exponentials.  Crossing $1$, you should integrate: $\int_{1^-}^{1^+}y''+3y'=1=y'+3y|_{1^-}^{1^+}$  As $y$ can't change instantaneously, $y'$ has to go from $0$ to $1$.  Then solve it starting at $t=1$ with $y=0, y'=1$