I want to know how to solve the above equation. x is defined in [0,1] and $\alpha$ is a constant. Wolfram Alpha already gives me a solution. I'm trying to solve using Frobenius method, assuming a solution $y(x) = \sum_{n=0}^{\infty}a_{n} x^{r+n} $. I obtained the solution to the indicial equation: $\alpha_{\pm} = \frac{1}{2}(1+\sqrt{1+4\alpha})$ and the recursion formula for the coefficients (for $\alpha_{+})$: \begin{equation} a_{n} = \frac{-i i^n\left( \sqrt{4\alpha+1}+3/2 \right)_{n-1}}{\Gamma(n+1)\left( \sqrt{4\alpha+1}+2 \right)_{n-1}}a_{0}, \end{equation} where $()_{n-1}$ is the Pochhammer symbol. My problem is that I don't know how to write this recursion in terms of Bessel functions. Maybe I should apply first a smart change of variables and write the equation as a Bessel-like one ? Any help is welcome.

  • $\begingroup$ @xpaul, that question isn't very helpful, the equation discussed there differs in the first derivative. $\endgroup$ – avemaria Jul 26 '17 at 17:01
  • $\begingroup$ @xpaul, it's not a matter of the imaginary unit, but I have a term $x^2$ and they have a term $x$. $\endgroup$ – avemaria Jul 26 '17 at 17:10
  • $\begingroup$ @xpaul, applying your change of variable I obtain the equation $t^{2}y''(t)-t^{2}y'(t)-\alpha y(t) = 0$, which is not the same in the link you posted. $\endgroup$ – avemaria Jul 26 '17 at 17:20
  • $\begingroup$ I made a mistake. $\endgroup$ – xpaul Jul 26 '17 at 20:35

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