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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

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

, 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.