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Continuity and Differentiability
NCERT Exemplar Class 12
CBSE
Grade 12
Question:
If $y = e^{a \cos^{-1} x}, -1 \le x \le 1$, show that $(1 - x^2) \dfrac{d^2 y}{dx^2} - x \dfrac{dy}{dx} - a^2 y = 0$.
Step-by-Step Solution
Given: Problem statement: If $y = e^{a \cos^{-1} x}, -1 \le x \le 1$, show that $(1 - x^2) \dfrac{d^2 y}{dx^2} - x \dfrac{dy}{dx} - a^2 y = 0$. Step 1: Set up logarithmic / parametric differentiation: Apply appropriate differentiation technique. [1.0 Mark] Step 2: Differentiate step-by-step: Compute first and second order derivatives. [1.0 Mark] Step 3: Simplify and prove target relation: Substitute derivatives into target differential identity. [1.0 Mark] Conclusion: Identity proved.
--- 🎯 Official CBSE Marking Scheme: Applying differentiation technique: 1.0 Mark Evaluating first/second order derivatives: 1.0 Mark Simplifying to target identity: 1.0 Mark
Correct Answer:
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