I am facing with this equation:
$(x^2−4a)y''+xy'−(x^2+b^2)y=0$
where $a$ and $b$ are constants. We know if the term $−4ay''$ does not exist, the solution would be modified bessel function.
the question is: is there any solution for this equation (except Frobenius)? or is there any substitution (change of variable) to change the equation to bessel equation?
I hope somebody can help me out.
Thanks in advance!
Consider the differential equation \begin{align} (x^2 - 4 a) \, y'' + x \, y' - (x^2 + b^2) \, y = 0. \end{align} Make the transformation $x = t -2a$ to obtain the equation \begin{align} t \left(1 - \frac{t}{4a} \right) \, y'' + \frac{1}{2} \, \left( 1 - \frac{t}{2a}\right) \, y' + \frac{1}{4a} \, ((t-2a)^{2} + b^{2}) \, y = 0. \end{align} Now make the substitution $t = 4 a u$ to obtain the equation \begin{align} u(1-u) y'' + \left(\frac{1}{2} - u\right) y' + (b^{2} + 4 a^{2} - 16 a^{2} u + 16 a^{2} u^{2})y=0. \end{align} Comparing this differential equation to that of the generalized spheroidal wave equation, obtained from L. J. El-Jaick, given by \begin{align} z(z-1) w'' ( B_{1} + B_{2} z) w' + (B_{3} - 2 \eta \omega (z-1) + \omega^{2} \, z (z-1))w = 0 \end{align} then the proposed differential equation is a member of the generalized spheroidal wave equation class of equations.