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If x = sin(\(\frac{1}{a}\)log y) , show that (1–x2)y2–xy1–a2 y = 0

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 Note: y2 represents second order derivative i.e.\(\frac{d^2y}{dx^2}\) and y1 = dy/dx

Given,

 x = sin(\(\frac{1}{a}\)log y) 

\((logy)=asin^{-1}x\)

y = \(e^{asin^{-1}x}\)...............equation 1

to prove : (1 - x2)y2 -xy1 - a2=0

We notice a second–order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find \(\frac{d^2y}{dx^2}\)

As, \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)

So, lets first find dy/dx

\(\because\) y = \(e^{asin^{-1}x}\)

And y = et

Again differentiating with respect to x applying product rule:

Using chain rule and equation 2:

Using equation 1 and equation 2 :

∴ (1–x2)y2–xy1–a2y = 0……proved

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