Converting 2nd order ODE to Bessel Function

[rjg6]

Homework Statement

I am attempting to solve the 2nd order ODE as follows using the generalized solution to the Bessel's equation

Homework Equations

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

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.

 

[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