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Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #97

For y=\(\left(x + \sqrt{x^{2} + 1}\right)^{n}\), prove \(x^{2} + 1\)y″ + \(x\)y′ + \(- n^{2}\)y = \(0\). Assume all open intervals where the displayed expressions are real and differentiable.

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. The positive base x+√(1+x²) equals exp(asinh x). Logarithmic differentiation gives y′=ny/√(1+x²).
  2. Multiply by √(1+x²): √(1+x²)y′=ny.
  3. Differentiate and multiply again by √(1+x²): (1+x²)y″+xy′=n√(1+x²)y′=n²y.
  4. Therefore (1+x²)y″+xy′=n²y, proving the identity.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The positive base x+√(1+x²) equals exp(asinh x). Logarithmic differentiation gives y′=ny/√(1+x²).
  • [1 mark] Multiply by √(1+x²): √(1+x²)y′=ny.
  • [1 mark] Differentiate and multiply again by √(1+x²): (1+x²)y″+xy′=n√(1+x²)y′=n²y.
  • [0.5 marks] Therefore (1+x²)y″+xy′=n²y, proving the identity.

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