## Converting 2nd order ODE to Bessel Function

1. The problem statement, all variables and given/known data
I am attempting to solve the 2nd order ODE as follows using the generalized solution to the Bessel's equation

2. Relevant equations
original ODE:
x$$d^{2}y/dx^{2}$$-3$$dy/dx$$+xy=0

3. The attempt at a solution
My first thought is to bring out an x^-1 outside of the function so that I end up with:
$$x^{-1}($$$$x^{2}$$$$d^{2}y/dx^{2}$$-3x$$dy/dx$$+$$x^{2}$$y)=0
I would then solve the resulting Bessel equation found inside the parentheses, and multiply the resulting solution by x^-1. Is this at all a legal operation? Thank you.
 Recognitions: Gold Member Science Advisor Staff Emeritus There is no need for the x-1 outside. Just multiply both sides of the original equation by x.
 Now if the opposite were true and I was trying to drop the power of x's by 1: example: (x^3)$$d^{2}y$$/$$dx^{2}$$+(x^2)dy/dx+(x^3)y=0 Could I then instead divide by x to come up with: (x^2)$$d^{2}y$$/$$dx^{2}$$+(x)dy/dx+(x^2)y=0 with the understanding that the solution to the resulting Bessel function would exclude any results for when x-> 0?

Recognitions:
Gold Member