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Bessel Function, proofs

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USING THE BESSEL FUNCTION OF ORDER ZERO:

Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

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Solution Summary

This solution is comprised of a detailed explanation to verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

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Problem:

Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.

Solution:

The general form of a Bessel equation is:
( 1)
and its general solution is
( 2)
where Bessel's function of first species of order "".
(Actually this is ...

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