How to solve the following ordinary differential equation \begin{equation} f_{xx}^{^{\prime \prime }}+2axf_{x}^{^{\prime }}-2sf=0 \end{equation} with $a$ and $s$ are arbitrary real numbers and the boundary conditions \begin{equation} f\left( \delta \right) =f\left( -\delta \right) =1 \end{equation}
After checking some books, I got the general solution \begin{equation} f\left( x\right) =C_{1}\Phi \left( -\frac{s}{2a},\frac{1}{2},-ax^{2}\right) +C_{2}\Psi \left( -\frac{s}{2a},\frac{1}{2},-ax^{2}\right) \end{equation} where $\Phi$ and $\Psi$ are degenerate hypergeometric functions. Are there any other forms of the solution, such as double integral or serious?