# Solution to simple inhomogeneous differential equation

What's the solution to this differential equation?

$\frac{\mathrm{d}u(t)}{\mathrm{d}t}+u(t)=\sum_i\delta(t-t_i)$

Intuitively I would say it's

$u(t)=\sum_i\mathrm{e}^{t_i-t}\sigma(t-t_i)$

where $\sigma(t)$ is the heaviside step function... is that correct? Thanks.

Close; Taking the Laplace transform gives you $$sU(s)-u(0) + U(s) =\sum_i e^{-t_is} \implies U(t) = \frac{u(0)}{s+1} + \sum_i\frac{e^{-t_is}}{s+1}$$ Taking the inverse Laplace transform gives $$u(t) = u(0)e^{-t}+ \sum_i e^{-(t-t_i)}\sigma(t-t_i)$$ which is the same, just with the homogenous solution in there.