Question
Change the independent variable from $x$ to $\theta$ by $x=\cos \theta$ and show that the Legendre equation$$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0$$becomes$$\frac{d^{2} y}{d \theta^{2}}+\cot \theta \frac{d y}{d \theta}+2 y=0.$$
Step 1
We need to express derivatives with respect to \( x \) in terms of derivatives with respect to \( \theta \). Show more…
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Change the independent variable from $x$ to $\theta$ by $x=\cos \theta$ and show that the Legendre equation $$ \left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0 $$ becomes $$ \frac{d^{2} y}{d 0^{2}}+\cot \theta \frac{d y}{d \theta}+2 y=0 $$
PARTIAL DIFFERENTIATION
Change of variables
The equation for the associated Legendre functions (and for Legendre functions when $m=0$ ) usually arises in the form (see, for example, Chapter 13, Section 7 ) $$ \frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d y}{d \theta}\right)+\left[n(l+1)-\frac{m^{2}}{\sin ^{2} \theta}\right] y=0 $$ Make the change of variable $x=\cos \theta$, and obtain $(10.1)$.
SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS; LEGENDRE POLYNOMIALS; BESSEL FUNCTIONS; SETS OF ORTHOGONAL FUNCTIONS
The associated Legendre functions
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