# Linear Differential Equation to solve?

1. Jan 18, 2012

### abrowaqas

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

this is Cauchy's LDE.. someone help me to solve this equation.

x^2 y'' + 3x y' + y = 1/(1-x)^2

2. Relevant equations

3. The attempt at a solution

i started it with substituting
x= e^t
then ln x = t
and d/dt = D

hence the equation becomes

{ D(D-1) +3D +1} y = 1/(1+e^t)^2

and i got characteristics equation
as
Yc=(c1 + c2x ) e^-1

now i have problem in findind Particular Intergral i-e Yp..

i-e
Yp = 1 / {(D+1)^2 (1+e^t)^2 } ???

somebody help to complete its solution?

Last edited: Jan 18, 2012
2. Jan 18, 2012

### obafgkmrns

I've only seen that sort of equation referred to as an Euler's differential equation. See, e.g., http://mathworld.wolfram.com/EulerDifferentialEquation.html.

You solve the homogeneous equation by substituting y = x^n and finding appropriate values for n. (In your case, you run into a double root, so you'll have to use reduction of order to find the second solution.) Then you can use variation of parameters to find the solution to the nonhomogeneous equation.

3. Jan 19, 2012

### abrowaqas

thanks Obafgkmrns.... i got it..