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$$\frac{d^{2} y}{d 0^{2}}+\cot \theta \frac{d y}{d \theta}+2 y=0$$
Step 1
Given that $x = \cos \theta$, we can find that $\theta = \cos^{-1} x$ and $\frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{d\theta}{dx}$. 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 $$\frac{d^{2} y}{d \theta^{2}}+\cot \theta \frac{d y}{d \theta}+2 y=0.$$
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