In this series of posts about Legendre differential equation, I would like to de-construct the differential equation down to the very bones. The motivation for this series is to put all that I know about the LDE in one place and also maybe help someone as a result.

The Legendre differential equation is the following:

where and

We will find solutions for this differential equation using the power series expansion i.e

We will plug in these expressions for the derivatives into the differential equation.

– (i)

– (ii)

– (iii)

** Note: Begin

Let’s take .

As n -> . , -> .

As n -> , -> .

Again performing a change of variables from to n.

** Note: End

(iii) can now be written as follows.

– (iv)

(i)+(ii)+(iv).

x = 0 is a trivial solution and therefore we get the indicial equation:

We get the following recursion relation on the coefficients of the power series expansion.

**Next post**: *What do these coefficients mean ?*

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