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I have updated the information. Truly I DON'T UNDERSTAND WHAT THE PROFESSOR HAS ASKED FOR? THAT IS WHY I NEED HELP. BELOW IS WHAT HE ASKED IN ADDITION TO MY PREVIOUS ANSWER.
On this problem, we know from a previous problem that the Legendre
polynomials satisfy the DE. It is a second order DE. Usually these
have two linearly independent solutions. Are these the only polynomials
that satisfy the DE, or is there another set, linearly independent of
the ones we found?
Legendre's differential equation, i.e. , (1-x^2)y''(x) - 2xy'(x) + n(n+1)y(x)=0. Find all solutions to Legendre's differential equation assuming solutions of the form y(x) = Pr(x). Pr(x) is considered the same as Pn(x).
Solutions to Legendre's Differential Equation are investigated.