Class 12 Differential Equations · Very Long Answer5 Marksseparable_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #112
Solve x(1 + y²)dx − y(1 + x²)dy = 0, y(0) = 1.
✅ Step-by-Step Solution
Final Answer: y = √(1 + 2x²), x ∈ ℝ.
- Separate x dx/(1 + x²) = y dy/(1 + y²).
- Integrate ½ln(1 + x²) = ½ln(1 + y²) + C.
- Hence 1 + y² = A(1 + x²), A > 0.
- At (0, 1), A = 2, so y² = 1 + 2x².
- The initial positive sign selects y = √(1 + 2x²), x ∈ ℝ.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Separate x dx/(1 + x²) = y dy/(1 + y²).
- [1 mark] Integrate ½ln(1 + x²) = ½ln(1 + y²) + C.
- [1 mark] Hence 1 + y² = A(1 + x²), A > 0.
- [1 mark] At (0, 1), A = 2, so y² = 1 + 2x².
- [1 mark] The initial positive sign selects y = √(1 + 2x²), x ∈ ℝ.
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