Second-order nonlinear ODE with Dirac Delta Can anyone help me with the following differential equation?
$$
2x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0,
$$
where $\delta$ represents the Dirac Delta.
I tried Mathematica but with no luck. And I really don't know where to start... Can anyone help me?
Note: I first thought that such an equation wouldn't be solvable but If I search for a solution of $x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0,$ mathematica can actually find a solution so maybe the above equation can also be solved.
 A: Dividing the whole equation by $x(t) x'(t)$, we have
$$
2 \frac{x''(t)}{x'(t)} - \frac{x'(t)}{x(t)} + k \frac{\delta(x-a) x(t)}{x'(t)} = 0
$$
hence
$$
\frac{d}{dt}\log\left\{\frac{x'(t)^2}{x(t)}\right\} + k \frac{\delta(x-a) x(t)}{x'(t)} = 0
$$
If you integrate the equation around $a$, you have
$$
\int_{a-\epsilon}^{a+\epsilon} \left(\frac{d}{dt}\log\left\{\frac{x'(t)^2}{x(t)}\right\} + k \frac{\delta(x-a) x(t)}{x'(t)}\right)dt = \log\left\{\frac{x'(t)^2}{x(t)}\right\}_{a-\epsilon}^{a+\epsilon} + k \frac{x(a)}{x'(a)}
$$
Taking the limit as $\epsilon \to 0$,
$$
\left[\log\left\{\frac{x'(t)^2}{x(t)}\right\}\right]_{t=a} + k \frac{x(a)}{x'(a)} = 0,
$$
where
$$
[f(t)]_{t=a} = \lim_{t \to a^+} f(t) - \lim_{t \to a^-} f(t)
$$
is the jump condition in $t=a$.
Now, you have the problem
$$
\frac{d}{dt}\log\left\{\frac{x'(t)^2}{x(t)}\right\} = 0 \qquad \begin{cases}t < a\\ t > a\end{cases}
$$
with 
$$
\left[\log\left\{\frac{x'(t)^2}{x(t)}\right\}\right]_{t=a} + k \frac{x(a)}{x'(a)} = 0,
$$
which can be easily integrated.
