Hello all, I seem to be stuck trying to prove this one, I'm not sure if I'm headed down the right path or just missing something obvious(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

The equation x^{2}y''+xy'+(κ^{2}x^{2}-p^{2})y=0, where κ constant, appears frequently in applications. Prove that J(κx) is a solution by changing the form of Bessel's equation of order p. Do this by letting [itex]\widetilde{x}[/itex]=κx in the above equation and apply the chain rule.

2. Relevant equations

Well the first solution of Bessel's equation is

J_{p}(x)=[itex]\sum^{\infty}_{k=0}\frac{(-1)^{k}}{k!\Gamma(k+p+1)}(\frac{x}{2})^{2k+p}[/itex]

3. The attempt at a solution

I believe the method I should be using is to use y(x)=J_{p}(κx)=J_{p}([itex]\widetilde{x}[/itex]) but from there I can derive y'=[itex]\sum^{\infty}_{k=1}\frac{(-1)^{k}(2k+p)}{k!\Gamma(k+p+1)}(\frac{\widetilde{x}}{2})^{2k+p-1}[/itex] and y''=[itex]\sum^{\infty}_{k=2}\frac{(-1)^{k}(2k+p)(2k+p-1)}{k!\Gamma(k+p+1)}(\frac{\widetilde{x}}{2})^{2k+p-2}[/itex] but from there I get fuzzy, I can't replace [itex]\widetilde{x}[/itex]=κx in a way that works out.

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# Bessels Equation J(κx) instead of J(x)

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