General solution of x²y'' + xy' + (x² - 1/4)y = 0

[owlman76]

Find the general solution of

x²y'' + xy' + (x² - 1/4)y = 0

and express it in terms of trigonometric functions. (You don't actually have to solve the equation from a trial function in this problem, but you must identify the differential equation. Then, you may write the solution and prove that it can be written as trigonometric functions.)

 

[coelho]

The solutions of this are called Bessel Functions.

http://en.wikipedia.org/wiki/Bessel_function