I am trying to understand a physical system and have arrived at the following equation:
$$\mathcal{S} = \int_{z = -\infty}^{z = \infty} dz \left\lbrace f_\rho[\rho] + \dfrac{m}{2} \bigg| \dfrac{\partial \rho}{\partial z} \bigg| ^2 + \dfrac{\rho ^2 (z) \delta(z-z_0)}{2}\right\rbrace$$
where I seek the $ \rho (z) $ that minimizes the value of $\mathcal{S}$. $ \rho (z) $ is a function of $ z $, with $ \rho ' = d\rho / dz$. $ f [\rho] = a \rho ^2 + b \rho ^3 + c \rho ^4$ is a functional of $ \rho (z) $, and $ f_\rho[\rho] = \frac{df}{d\rho} $.
I thus write an Euler-Lagrange equation:
$$ \dfrac{\partial \mathcal{S}}{\partial \rho} = \dfrac{\partial}{\partial z} \left( \dfrac{\partial \mathcal{S}}{\partial \rho '}\right)$$
I have been unable to solve this equation for $ {\rho(z)} $, and was wondering how to do so.
I have managed to simplify the EL equation to:
$$ \dfrac{\partial f_\rho [\rho]}{\partial \rho} + \rho \delta(z-z_0) = m \dfrac{\partial ^2 \rho}{\partial z^2} $$
but I don't know where to go from here. Any help would be much appreciated.