I need to solve the following equation
$$ \left(v\frac{\partial L}{\partial v}-L\right)^{2}=\left(\frac{\partial L}{\partial v}\right)^{2}c^{2}+\frac{1}{M^{2}}\left(\frac{\partial L}{\partial v}\right)^{4}+m^{2}c^{4} $$
for $L=L(q,v)$, where $c$, $M$, and $m$ are constants, $M>>m$.
I tried to use a Fourier transform: (let $F_{v}[L](\omega)=F$):
$$ F_{v}\left[\frac{\partial L}{\partial v}\right](\omega)=-i\omega F_{v}[L]=-i\omega F $$
$$ F_{v}[v](\omega)=-i\sqrt{2\pi}\delta'(\omega) $$
Here $\delta'(\omega)$ is the derivative of the delta function. Substituting these expressions in the given equation, we set $t=F^{2}$, to obtain the quadratic equation
$$ c^4 m^2-t \left(c^2 \omega ^2-\omega\sqrt{2\pi} \delta (\omega )'-1\right)+\frac{t^2\omega^4}{M^2}=0 $$
At this point it seemed that all is well, but Mathematica provides a cumbersome expression for the roots and can't do the inverse Fourier transform.
Can you help me to solve this equation? Maybe there are more artful ways?
