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I have encountered Bessel functions in the context of Partial Differential Equations. It is supposed that Bessel functions are also the solutions of the Ordinary differential Equation: $$x^2y''+xy'+(x^2-\alpha^2)y=0$$ But I don't know how to deduce from this equation the Bessel functions. Of course I could just prove that they satisfy the equation therefore they are solutions, but I'm interested in seeing how they can be obtained by solving de ODE.

Thanks a lot

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I was going to post a comment but it turned out to be too long.

I am slightly confused what you are asking. This particular ODE clearly has no analytic solution in term of the functions we are familiar with, and since it is 2nd order, the rule of thumb is that there are 2 'independent' solutions. They are Bessel functions.

The link in the comment shows the Frobenius method, which gives you a series representation of Bessel functions. This is one way how Bessel functions can the defined. (I was going to say, this is how Bessel function is defined but I am not 100% sure)

The intuition is that you try a series solution like the Taylor series, and you obtain a recurrence relation which gives you the coefficient of each term.

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