Transforming a Differential Equation to a Legendre's Equation

Question

Am trying to transform the following DE into a Legendre's equation, but then i don't have sufficient knowledge on how to go about it. I tried following some tutorials but then their explanation is not 'satisfactory' Differential Equation.

Answer 1

Legendre's equation in a particular form (essentially the equation you provided in an image ) is $$ y'' + \cot \theta \,y' + \left[n(n+1)\right] y = 0 \qquad(1) $$ where the $'$ indicates differentiation of $y(\theta)$ wrt $\theta$.

Let $z=\cos \theta$ and consider $y(\theta(z))$

Use the chain rule to derive to the derivatives $y'$ and $y''$.

e.g. $$ \frac{dy}{dz} = \frac{dy}{d \theta} \frac{d \theta}{d z} $$ They should be $$ \frac{dy}{d \theta} = -\sin \theta \frac{dy}{d z} $$

and $$ \frac{d^2y}{d {\theta}^2}=(1-z^2)\frac{d^2y}{d z^2}-z\frac{dy}{d z} $$ Substitute in $(1)$ to get the more common form of Legendre's differential equation $$ (1-z^2)\frac{d^2y}{d z^2}-2z\frac{dy}{d z}+\left[n(n+1)\right]y = 0 $$