Question:
If $y = e^{a \sin^{-1} x}$, then $(1 - x^2) y_2 - x y_1 =$
If $y = e^{a \sin^{-1} x}$, then $(1 - x^2) y_2 - x y_1 =$
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For functions of the form $y = e^{a f(x)}$, squaring and cross-multiplying the first derivative helps to eliminate the radical term before doing the second derivative.
Updated On: May 31, 2026
- $a^2 y$
- $-a^2 y$
- $a y$
- $-a y$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
We use successive differentiation and the chain rule of differentiation to find the first derivative $y_1$ and second derivative $y_2$ of the given function.
Step 2: Meaning
Differentiating $y$ once gives us an expression containing $\sqrt{1-x^2}$. Squaring and differentiating again will lead us directly to the required differential equation.
Step 3: Analysis
Given $y = e^{a \sin^{-1} x}$. Differentiating once with respect to $x$: \[ y_1 = e^{a \sin^{-1} x} \cdot \frac{a}{\sqrt{1 - x^2}} = \frac{ay}{\sqrt{1 - x^2}} \] Cross-multiplying: \[ y_1 \sqrt{1 - x^2} = ay \] Squaring both sides: \[ y_1^2 (1 - x^2) = a^2 y^2 \] Differentiating both sides with respect to $x$ using the product rule and chain rule: \[ 2y_1 y_2 (1 - x^2) + y_1^2 (-2x) = a^2 (2y y_1) \] Dividing both sides by $2y_1$ (since $y_1 \neq 0$): \[ (1 - x^2) y_2 - x y_1 = a^2 y \]
Step 4: Conclusion
The value of $(1 - x^2) y_2 - x y_1$ is $a^2 y$. Final Answer: (A)
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