Solve the differential equation $(1-x^2)y”-2xy’=\sum_{n=1}^{\infty}P_n(x)$

ordinary differential equationsspecial functions

So I was reading back over my course problem sheet for differential equations ( I'm studying for exams right now) and I came across this question:

Given that

$$\tfrac{2}{(5-4x)^{1/2}}=\sum_{n=0}^{\infty}\tfrac{1}{2^n}P_n(x)$$

For $|x|\leq1$

Find a particular solution of the equation

$$(1-x^2)y''-2xy'=\tfrac{2}{(5-4x)^{1/2}} -1$$

I've been wracking my brains trying to solve it but I can't figure out how. From my research I found that one way to solve a second order linear inhomogenous differential equation ( as this one is ) is to use the method of variation of parameters. However my teacher assured me that this could be solved using only material covered in the notes and this method is not in them.

Here's what we covered up to the point of the notes that corresponded to this question. He also said that we should be able to solve any question in the course without referring back to our first course in solving o.d.e.'s which thought us methods such as froebenius, characteristic equations, integrating factor method etc ….

  1) Linear independence of functions

     a) Criteria for checking linear independence

     b)Finding a second solution 


  2) Orthogonal function expansions (Sturm-Liouville theorem)


    a) Self adjoint form

    b) linear independence  

    c)Orthogonality 

    d)Completeness

So my question is how can we use the concepts I've listed to solve this equation , or is it even possible to do so ? Even though he said it was it seems like there's just a few ingredients missing!

Note:just to be clear this isn't a homework question it's just from a problem sheet i'm studying for my exam in a few weeks.

Best Answer

The $P_n(x)$'s are the Legendre polynomials, which satisfy $$ (1 - x^2)P_n''(x) - 2xP_n'(x) = -n(n+1)P_n(x).$$

Therefore, if $f(x)$ is the function $$ f(x) = \sum_{n = 0}^\infty a_n P_n(x),$$ then the solution to the differential equation $$ (1 - x^2)y'' - 2xy' = f(x)$$ must be $$ y=- \sum_{n=0}^\infty \frac{a_n}{n(n+1)}P_n(x).$$

Your question is a special case, with $a_n = 1 / 2^{n} - \delta_{n, 0}$.

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