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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