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Class 12 Differential Equations · Long Answer3 Marksseparable_differential_equations

Class 12 Maths · Differential Equations · Long Answer #63

Solve √(1 + x² + y² + x²y²) + xy y′ = 0.

✅ Step-by-Step Solution

Final Answer: √(1 + y²) + √(1 + x²) + ln|x/(1 + √(1 + x²))| = C, on regular branches xy ≠ 0.
  1. Factor the radical as √(1 + x²)√(1 + y²). The original equation cannot hold at x = 0 or y = 0 for a finite derivative.
  2. Separate y dy/√(1 + y²) = −√(1 + x²)dx/x.
  3. The left integral is √(1 + y²). On the right, put u = √(1 + x²); ∫√(1 + x²)dx/x = u + ½ln|(u − 1)/(u + 1)| = u + ln|x/(u + 1)|.
  4. Therefore √(1 + y²) + √(1 + x²) + ln|x/(1 + √(1 + x²))| = C on connected regular branches.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Factor the radical as √(1 + x²)√(1 + y²). The original equation cannot hold at x = 0 or y = 0 for a finite derivative.
  • [1 mark] Separate y dy/√(1 + y²) = −√(1 + x²)dx/x.
  • [1 mark] The left integral is √(1 + y²). On the right, put u = √(1 + x²); ∫√(1 + x²)dx/x = u + ½ln|(u − 1)/(u + 1)| = u + ln|x/(u + 1)|.
  • [0.5 marks] Therefore √(1 + y²) + √(1 + x²) + ln|x/(1 + √(1 + x²))| = C on connected regular branches.

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