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

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

  • $\begingroup$ it's quite helpful.Thanks a lot. $\endgroup$ – ANJAN DASGUPTA Oct 15 '18 at 4:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.