Solution of Bessel Differential Equation Using Bessel Function

[Kopernikus89]

Hello
I have the following problem:
I must show that the Bessel function of order $$n\in Z$$

$$J_n(x)=\int_{-\pi}^\pi e^{ix\sin\vartheta}e^{-in\vartheta}\mathrm{d}\vartheta$$

is a solution of the Bessel differential equation

$$x^2\frac{d^2f}{dx^2}+x\frac{df}{dx}+(x^2-n^2)f=0$$

Would be very thankful for some help :-)

 

[Char. Limit]

Try differentiating it. I believe you can use differentiation under the integral sign to help you. Remember that a(x) and b(x) are constants here.

 

[Kopernikus89]

well the first 2 summands equal 0 (i hope I've calculated this correctly) but its more a problem with the third one. how can i show that this will also become 0?

 

[g_edgar]

Let's call your right-hand-side $F(x)$
Then: what do you get for $F'(x)$ and $F''(x)$

 

[ogirald]

Hello,
I'd like to know how to solve the ODE shown in the attached file using Bessel functions

I will be very grateful!