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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?

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    Dear ben, you can use the formula given in this [answer](http://math.stackexchange.com/questions/58234/how-to-solve-this-ode/58240#58240).2011-10-07

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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

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    @ChristopherTurnbull: Yes, away from $1$ you are back to the homogeneous case, but the initial conditions are different. We had $y'(0)=0$, but $y'(1^+)=1$. That starts a damped oscillation.2016-11-21