Legendre Differential Equation(#3): A friendly introduction

This post is just a note on the notation that is used across internet sources and books while referring to the LDE.

(1-x^2)y^{''} - 2xy^{'} + l(l+1) y = 0

If one takes p(x) = 1-x^2 , then it follows that p^{'}(x) = -2x . The differential equation can be rewritten as follows:

p(x)y^{''} + p^{'}(x)y^{'} + l(l+1) y = 0

Now the first two terms must seem familiar to you from the chain rule. ( (py)^{'} = py^{'} + yp^{'} ). Ergo,

(py^{'})^{'} + l(l+1)y = 0

or

\frac{d}{dx}(p \frac{dy}{dx}) + l(l+1)y = 0

Now, putting back the value of p :

\frac{d}{dx}\left((1-x^2) \frac{dy}{dx} \right) + l(l+1)y = 0

And you will see this form of the LDE also in many places and I thought it was worth mentioning how one ended up in that form.

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