I consider the ordinary differential equation of the form: \begin{equation} y''(x) + a y'(x) + (b e^{\lambda x} + c) y(x) = 0. \end{equation}
In the book 'Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems' (by Andrei D. Polyanin and Valentin F. Zaitsev) on the page 585/1487 there is written a solution to this equation of the form: $$ y=e^{−ax/2}[C_1 J_\nu(2\lambda^{−1}\sqrt{b} e^{\lambda x/2}) + C_2 Y_\nu(2\lambda^{−1}\sqrt{b} e^{\lambda x/2})], $$ where $\nu = \lambda^{−1}\sqrt{a^2 − 4c}$ and $J_\nu(z)$ and $Y_\nu(z)$ are Bessel functions.
The form of this soluton it's not too complicated, so I think that the way of solving this ODE is also not very difficult.
Can anyone suggest me how to derive this solution?