Mathematica cannot solve this system of ODEs, even after the decoupling suggested by user LeechLattice, even with $\alpha=\beta=0$ and $A=1$. This means that the existence of an explicit solution is highly unlikely. 

Here is the image of the corresponding Mathematica notebook: 

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/iVTVx.png