I'm trying to understand a linear second order differential equation with variable coefficients:
$\frac{\partial^2 u}{\partial z^2} = z (1 + z u)$
I already have the solution in an integral form
$u(z) = -\frac{z}{2} \int_0^{\pi/2} d\theta\: \exp(-\frac{z^2}{2}\cos\theta) \sin^{1/2}\theta$.
My question is: how do I find this solution? What technique/method to use?
Bonus question: What if I add an extra term $z\frac{\partial u}{\partial z}$ to the equation, how can I then solve this equation? The modified ode ($C$ is a constant) is
$\frac{\partial^2 u}{\partial z^2} = z (1 + z u + C \frac{\partial u}{\partial z})$