Differential equation involving the Dirac delta

Question

I have been trying to figure this out for a while, and I was wondering if anyone had any ideas. I need to solve the following differential equation:

$m\frac{d^2 r}{dt^2}=\epsilon\delta'(r)$,

where $\delta(r)$ is the Dirac delta function, and $m,\epsilon$ are constants. What would be the best way to go about this?

Cheers

Answer 1

My approach is based on two facts:

1. $\delta $ is the distribitional derivative of the Heaviside function $H = \chi_{[0,\infty)}$.
2. A distribution whose first derivative is identically $0$ is constant.

Thus, $mr'=\epsilon \delta +C = \epsilon H'+C$. By the same logic, $mr = \epsilon H +Ct+B$.

Answer 2

I have tried a little can't sure correct or not, $$m\frac{d^2r}{dt^2}=\epsilon\delta'(r)$$ $$\frac{d}{dt}\left[ m\frac{dr}{dt} -\epsilon\delta \right]=0$$ $$\left[m\frac{dr}{dt} -\epsilon\delta \right]=C$$ $$\int dr-\frac{\epsilon}{m}\int\delta\,dr =\int C\, dt$$ $$r-\frac{\epsilon}{m}=Ct + C'$$