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NCERT Class 12 · Ch 5 Miscellaneous Exercise · Q22

NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 22

For y=e^(a cos⁻¹x), −1≤x≤1, show (1−x²)y″−xy′−a²y=0.

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: The identity holds for −1<x<1, where the derivatives exist.
  1. The first derivative is y′=−ay/√(1−x²).
  2. Differentiate again: y″=a²y/(1−x²)−axy/(1−x²)^(3/2).
  3. Multiplication by 1−x² and subtraction of xy′ cancels the terms with x, leaving a²y. Subtracting a²y gives zero.
  4. Although y is defined at ±1, this differential identity is justified on the open interval; generally the derivatives are singular at the endpoints.

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