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