Class 12 Differential Equations · Short Answer2 Marksseparable_differential_equations
Class 12 Maths · Differential Equations · Short Answer #40
Solve y′ + (1 + y²)/(2y) = 0.
✅ Step-by-Step Solution
Final Answer: ln(1 + y²) + x = C, on intervals y ≠ 0.
- Rearrange to 2y dy/(1 + y²) = −dx; the original equation excludes y = 0.
- Integrate using u = 1 + y²: ln(1 + y²) = −x + C.
- Thus ln(1 + y²) + x = C, equivalently y = ±√(Ae^(−x) − 1), A > 0, with Ae^(−x) > 1.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Rearrange to 2y dy/(1 + y²) = −dx; the original equation excludes y = 0.
- [1 mark] Integrate using u = 1 + y²: ln(1 + y²) = −x + C.
- [0.5 marks] Thus ln(1 + y²) + x = C, equivalently y = ±√(Ae^(−x) − 1), A > 0, with Ae^(−x) > 1.
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