I am reading a book that talks about series solutions of differential equations, and I couldn't seem to understand the following question:
Consider the differential equation
and use the assumption that
then find the recurrence relation for the infinite series to be a solution.
I am not entirely sure what the question means, especially about the recurrence relation. The only thing that I could get from the book is that the original equation is the Legendre differential equation. How should I approach this problem? Thanks.
Best Answer
I will make a similar exercise with a simpler equation. I hope that you see the point.
Consider $$xy+y'=0$$ with $$y=\sum_{k=0}^\infty c_nx^n$$ Then $$y'=\sum_{k=1}^\infty nc_nx^{n-1}$$ Now substitute: $$x\sum_{k=0}^\infty c_nx^n+\sum_{k=1}^\infty nc_nx^{n-1}=0$$ That is $$\sum_{k=1}^\infty c_{n-1}x^n+\sum_{k=0}^\infty (n+1)c_{n+1}x^n=0$$ For each $n\ge 1$ we have that $$c_{n-1}+(n+1)c_{n+1}=0$$ or $$c_{n+1}=-\frac{c_{n-1}}{n+1}$$ This would be the recurrence relation for this example.