# second order differential equation of special function

Bessel function $$J_n(x)$$ and $$Y_n(x)$$ obeys the following differential equation: $$x^2 y''(x)+x y'(x)+(x^2-n^2)y=0,$$ where superscript ' denotes differentiation with respect to $$x$$.

In general, equation of the form: $$x^2 y''(x)+ \alpha\,x y'(x)+(\beta\,x^2+\gamma)y=0\;(\alpha, \beta,\gamma: \text{constants})$$ has a solution which can be expressed in terms of Bessel functions as discussed in the link: http://mathworld.wolfram.com/BesselDifferentialEquation.html. Wondering if the following equation has a solution in terms of any such special functions?

$$x^2 y''(x)+ \alpha\,x y'(x)+(\beta\,x^2+\gamma x+\theta)y=0,$$ where $$\alpha, \beta,\gamma, \theta: \text{constants and reals}.$$

Change of the form $$y=x^k u$$, where $$k$$ is the solution of $$k^2+(\alpha-1)k+\theta=0$$ should give the equation of the form $$x u'' +(\alpha+2k)u'+(\beta x+\gamma)u=0$$ This is can be solved as described here: http://eqworld.ipmnet.ru/en/solutions/ode/ode0211.pdf