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CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Q345 Markslinear_differential_equations

CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Question 34 (Differential Equations)

Solve \(y+\frac{d}{dx}(xy)=x(\sin x+\ln x)\), for x>0.

✅ Step-by-Step Solution

Final Answer: y=[−x² cos x+2x sin x+2 cos x+(x³/3)ln x−x³/9+C]/x².
  1. Expand to x y′+2y=x(sin x+ln x). Divide by x and use the integrating factor x².
  2. Then d(x²y)/dx=x²(sin x+ln x). Integrate x²sin x twice by parts to get −x²cos x+2x sin x+2cos x.
  3. Integration by parts gives ∫x² ln x dx=(x³/3)ln x−x³/9. Add the primitives and divide by x².

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Expand to x y′+2y=x(sin x+ln x). Divide by x and use the integrating factor x².
  • [3 marks] Then d(x²y)/dx=x²(sin x+ln x). Integrate x²sin x twice by parts to get −x²cos x+2x sin x+2cos x.
  • [1 mark] Integration by parts gives ∫x² ln x dx=(x³/3)ln x−x³/9. Add the primitives and divide by x².

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All Questions in CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3

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