Question

y= sin^-¹ x show that (1-x^2) d^2y/dx^2 -dy/dx=0

Answer 1

$\textbf{Answer :}$

Given, y = sin⁻¹x

So, differentiating both sides with respect to x, we get

dy/dx = 1/√(1 - x²)

⇒ √(1 - x²) dy/dx = 1

Squaring both sides, we get

(1 - x²) (dy/dx)² = 1

Now, differentiating both sides with respect to x, we get

d/dx {(1 - x²) (dy/dx)²} = d/dx (1)

⇒ (1 - x²) d/dx {(dy/dx)²} + (dy/dx)² d/dx (1 - x²) = 0

⇒ {(1 - x²) × 2 × (dy/dx) × d²y/dx²} + {(dy/dx)² × (- 2x)} = 0

⇒ (1 - x²) d²y/dx² - dy/dx = 0

Hence, proved.

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