Solving a differential equation with the Dirac-Delta function without Laplace transformations

Question

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?

Answer 1

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 $y=0, y'=1$