The Legendre differential equation is the second order ordinary differential equation (ODE) which can be written as:

which when rearranged to:
is called Legendre differential equation of order
, where the quantity
is a constant.

where
is the Legendre operator:
![{\displaystyle L={d \over dx}[(1-x^{2}){d \over dx}]+l(l+1)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d34364604dcad9d6c0c11cfa5f1f279fd5990cdb)
In principle,
can be any number, but it is usually an integer.
We use the Frobenius method to solve the equation in the region
. We start by setting the parameter p in Frobenius method zero.
,
,
.
Substituting these terms into the original equation, one obtains
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Thus
,
and in general,
.
This series converges when
.
Therefore the series solution has to be cut by choosing:
.
The series cut in specific integers
and
produce polynomials called Legendre polynomials.
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