Question

 If equation, then show that equation. Hence using Leibniz's formula, find the value of (1 - x2)yn+2 - (2n + 1)xyn+1.

09 Jan 2026
Answer :
Word Count : 80
Given (y=e^{m\sin^{-1}x}). Differentiate with respect to (x): [ y_1=\frac{dy}{dx}=e^{m\sin^{-1}x}\cdot \frac{m}{\sqrt{1-x^2}}=\frac{m}{\sqrt{1-x^2}},y. ] Differentiate again: [ y_2=\frac{d}{dx}\left(\frac{m}{\sqrt{1-x^2}}y\right) =\frac{m}{\sqrt{1-x^2}}y_1+\frac{mx}{(1-x^2)^{3/2}}y. ] Substitute (y_1=\frac{m}{\sqrt{1-x^2}}y): [ ___ _____ ________ __________ ______ ______ _________ ____ ____ ____ _____ _________.
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