Solving ODE with variable coefficients

[rammohanRao]

Homework Statement

I wanted to solve a ode which has Brownian motion as a variable coefficient

Homework Equations

2x²y'' + y' -ρy = 0

where x is the Brownian motion with respect to time
ρ is a constant

The Attempt at a Solution

I have tried power series with no solution. Is there a solution to this. IS there any easy way to solve this ODE. Once this ode is tranformed I need to find the roots.

 

[rammohanRao]

Any hints

I have tried to reduce the order but could not.

Is there any transformation that I can apply. I tried y = x^r it did not work

Please guide me...

 

[LCKurtz]

Maple 13 gives a solution involving exponentials, first degree polynomials, and
Bessel functions multiplied together. This is also a special case of equation 17 at this link:
http://eqworld.ipmnet.ru/en/solutions/ode/ode-toc2.htm

Whether or not that will be helpful to you, I don't know.

 

[rammohanRao]

Thanks for the hints.

I saw the solution in maple15 which involves intergal and exponetials.Its little complex.
There is a tranformation required for this equation which I'm not able to get

Also it is not a special case of 17

Now in short
I need to know a transformation when you differentiate you get 1 and if you differentiate it again you get x^2