I'm trying to solve the following second order differential equation (which is supposed to be solvable analytically):

$\ddot y + 3\alpha \dot y +\beta^2 \exp\left(-2\alpha t\right) y=0$,

being $\alpha$ and $\beta$ constants and $y=y\left(t\right)$. The initial conditions ($t_i=0$) are set such that:

$y_{i}=\frac{\beta}{2\sqrt{\alpha \gamma}}\exp\left(i\beta/\alpha\right)$

$\dot y_i = - \frac{\beta}{2\sqrt{\gamma}}\exp\left(i\beta/\alpha\right)\left[\sqrt{\alpha}+i \frac{\beta}{\sqrt{\alpha}}\right]$

As you see, both are complex numbers. $\gamma$ is another constant.

I know that I can recast the former equation such that:

$\frac{\textrm{d}}{\textrm{d}t}\left[\exp\left(3\alpha t\right)\dot y\right]+\beta^2 \exp\left(\alpha t\right)y=0$,

which would correspond to a Sturm-Liouville equation. However, I don't what to do next.


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