Converting 2nd order ODE to Bessel Function

[rjg6]

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:
xd^{2}y/dx^{2}-3dy/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}-3xdy/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.

[HallsofIvy]

There is no need for the x-1 outside. Just multiply both sides of the original equation by x.

[rjg6]

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?

[HallsofIvy]

Well, because x2 still multiplying the second derivative that would be a problem any way, but you are right if you divided by something that completely got rid of a function multiplying the highest derivative, then you would have to add that condition.

[abhishekgoyal]

Hey I need some help in converting the second order differential equation into..
I was able to convert the original equation into the following form x^2*y''+2x*y'+x^2*y=0
I am not able to move forward from here..
Please could you suggest some method