# How do you combine Bessel functions?

## Main Question or Discussion Point

Hi,
I have been trying to solve this differential equation for a while now. Now I get to the point where I have the solution, but it includes an integral.

The integral is

$$\int x J_{1/4}(ax) J_{1/4}(bx) e^{-x^2t}dx$$

, where a and b are constants, and the integral is from zero to infinity. I think I can figure out how to integrate this by using a table of integral, but I need to only have one Bessel function in it.
How can I combine the two Bessel functions?

Any help is much appreciated.

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What do you mean by 'need to have one Bessel function in it?' If you mean the antiderivative, well I believe the antiderivative has no closed form. The problem is the e^(-t*x^2) in there.

Why don't you post the original DE?

thank you for replying. I thought there's a way to make the product of two Bessel function become one function, or square of one function.

But never mind, I found the solution to the integral.