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If y =etan–1x , Prove that: (1+x2)y2+(2x–1)y1=0

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

Given,

y = etan–1x……equation 1

to prove : (1–x2) y2–xy1–2=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

\(\frac{dy}{dx}=\frac{d}{dx}e^{tan^{-1}x}\) 

Using chain rule we will differentiate the above expression

And y = et

Again differentiating with respect to x applying product rule:

Using chain rule we will differentiate the above expression-

∴ (1+x2)y2+(2x–1)y1=0 ……proved

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