Linear Differential Equation to solve?

[abrowaqas]

Homework Statement

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

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

Homework Equations

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?

 

[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.

 

[abrowaqas]

thanks Obafgkmrns... i got it..