# Solving ODE with variable coefficients

1. Mar 8, 2012

### rammohanRao

1. The problem statement, all variables and given/known data

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

2. Relevant equations

2x2y'' + y' -ρy = 0

where x is the Brownian motion with respect to time
ρ is a constant
3. 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.

Last edited: Mar 8, 2012
2. Mar 8, 2012

### rammohanRao

Any hints

I have tried to reduce the order but could not.

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

3. Mar 8, 2012

### 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.

4. Mar 10, 2012

### 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