Non-Homogenous ODE Using Series Solution Method

[qwertyui123]

Homework Statement

I am studying some issues on the differential equation issues.
Currently I am involved with a non-homogeneous Chebyshev differential equation. The problem is to find power series solution to equation below:

(1-x^2)*y’’-x*y’+y=1/(1-x)

in which the x will be in [-1 ,1].
the main problem is the singularity of the non-homogenous part in x=1 which is included in problem domain.

in advanced thank a lot for contribution.

Homework Equations

The Attempt at a Solution

 

[lippyka]

The solution of the Chebyshev differential equation can be found by using the method of Frobenius. First, one must find the indicial equation. The indicial equation is determined by setting the coefficient of the highest power of x equal to zero, which produces the equation:r(r-1)-1 = 0 This indicial equation can be solved to obtain two roots, r1=1 and r2=2. This implies that the general solution for the Chebyshev differential equation will look like this:y(x) = c0 + c1x + c2x^2 + c3x^3+ ... + cnx^nThe coefficients of the terms in the solution can then be determined by plugging the equations into the original differential equation and solving for each coefficient. In order to deal with the singularity at x=1, we can use an appropriate transformation to remove the singularity before attempting to solve the equation. For example, if we define a new variable u = 1-x, then the equation can be rewritten as: (u^2)*y’’-(1-u)*y’+y=1/u Now, this equation can be solved using the same method outlined above.