Question

If \(y = (\sin^{-1}x)^2\), then \((1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} =\)

• A. 1
• B. 2
• C. \(\frac{1}{2}\)
• D. 4

Hint:

When dealing with inverse trigonometric functions, rearranging the first derivative equation (e.g., cross-multiplying the square root) before finding the second derivative often leads directly to the required differential equation.

Answer 1

The Correct Option is B

Step 1: First Derivative:
\(y = (\sin^{-1}x)^2\) \(\frac{dy}{dx} = 2\sin^{-1}x \cdot \frac{1}{\sqrt{1-x^2}}\) Rearrange to avoid quotient rule: \(\sqrt{1-x^2} \frac{dy}{dx} = 2\sin^{-1}x\)
Step 2: Differentiate Again:
Square both sides first to simplify (optional but helpful): \((1-x^2) \left(\frac{dy}{dx}\right)^2 = 4(\sin^{-1}x)^2 = 4y\) Differentiate w.r.t \(x\): \((1-x^2) \cdot 2\left(\frac{dy}{dx}\right) \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 (-2x) = 4 \frac{dy}{dx}\)
Step 3: Simplify:
Divide throughout by \(2\frac{dy}{dx}\) (assuming \(\frac{dy}{dx} \neq 0\)): \((1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 2\) Final Answer:
The value is 2.