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Legendre's Differential Equation

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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).

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Solution Preview

A solution is attached. It uses the series solution already worked out.

Taking the derivative of equation 4,

From equation 3, we see that . Substituting this in gives

Thus the Legendre polynomials satisfy Legendre's equation.

This is a previous problem and answer :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) = x^r.

The equation is .
The solution of the equation has the form , then we have
,
Then we have

We have the following cases:
Case 1: or , then , then we have , can be any real number, for all . This implies that is a constant.
Case 2: or , then , then , can be any real number, and we have for all . So . For , because , then ; for , we have . ...

Solution Summary

Solutions to Legendre's differential equation are investigated.

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