(a) If $y=e^{a \sin ^{-1} x}$, show that $\left(1-x^{2}\right) y_{n+2}-(2 n+1) x y_{n+1}-\left(n^{2}+a^{2}\right) y_{n}=0$.

Explanation

To show that $y=e^{a \sin ^{-1} x}$ satisfies the given differential equation, we need to find the higher order derivatives of y with respect to x and substitute them in the given equation. We have found the expressions for y1 and y2 and have used them to find the expression for yn.

(a) If $y=e^{a \sin ^{-1} x}$, show that $\left(1-x^{2}\right) y_{n+2}-(2 n+1) x y_{n+1}-\left(n^{2}+a^{2}\right) y_{n}=0$.

Explanation

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