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CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Q335 Marksparametric_derivatives

CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Question 33 (Continuity And Differentiability)

If x=sin t and y=sin(pt), prove (1−x²)y″−xy′+p²y=0 on regular branches.

✅ Step-by-Step Solution

Final Answer: (1−x²)y″−xy′+p²y=0.
  1. The chain rule gives dy/dt=y′ cos t.
  2. Differentiate again to get d²y/dt²=y″cos²t−y′sin t.
  3. Since d²y/dt²=−p²y, substitute cos²t=1−x² and sin t=x.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] The chain rule gives dy/dt=y′ cos t.
  • [3 marks] Differentiate again to get d²y/dt²=y″cos²t−y′sin t.
  • [1 mark] Since d²y/dt²=−p²y, substitute cos²t=1−x² and sin t=x.

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