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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}. $

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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

The change is described in https://www.crcpress.com/Handbook-of-Exact-Solutions-for-Ordinary-Differential-Equations/Zaitsev-Polyanin/p/book/9781584882978 (eq. 2.1.2.126 in russian edition)

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  • $\begingroup$ it's quite helpful.Thanks a lot. $\endgroup$ Commented Oct 15, 2018 at 4:56

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