Question

If y = (sin^–1 x)², prove that: (1–x²) y₂–xy₁–2=0

Answer 1

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

Given,

y = (sin^–1 x)² ……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 \(\frac{d^2y}{dx^2}\)

Using chain rule we will differentiate the above expression

Let t = sin^–1 x =>\(\frac{dt}{dx}=\frac{1}{\sqrt{1-x^2}}\) [using formula for derivative of sin^–1x]

And y = t²

 Again differentiating with respect to x applying product rule:

∴ (1–x²) y₂–xy₁–2=0 ……proved