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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #147

If x=tan((ln y)/a), with a≠0 and y>0 on a differentiable branch, prove (1+x²)y″+(2x−a)y′=0.

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. Differentiate x=tan((ln y)/a): 1=(1+x²)y′/(ay).
  2. Thus (1+x²)y′=ay. Differentiate this equality by the product rule.
  3. The result is (1+x²)y″+2xy′=ay′, so move the right side to the left.
  4. Therefore (1+x²)y″+(2x−a)y′=0.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate x=tan((ln y)/a): 1=(1+x²)y′/(ay).
  • [1 mark] Thus (1+x²)y′=ay. Differentiate this equality by the product rule.
  • [1 mark] The result is (1+x²)y″+2xy′=ay′, so move the right side to the left.
  • [0.5 marks] Therefore (1+x²)y″+(2x−a)y′=0.

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