Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #100
Solve y′ = xy/(x² + y²), y(0) = 1.
✅ Step-by-Step Solution
Final Answer: ln y − x²/(2y²) = 0, on the positive solution branch, x ∈ ℝ.
- For y > 0 treat x² as u(y): du/dy = 2u/y + 2y.
- Using IF 1/y² gives (u/y²)′ = 2/y, so x²/y² = 2ln y + C.
- At (0, 1), C = 0.
- Thus ln y − x²/(2y²) = 0, y ≥ 1. This defines a smooth even solution for every real x; its derivative at 0 is 0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For y > 0 treat x² as u(y): du/dy = 2u/y + 2y.
- [1 mark] Using IF 1/y² gives (u/y²)′ = 2/y, so x²/y² = 2ln y + C.
- [1 mark] At (0, 1), C = 0.
- [0.5 marks] Thus ln y − x²/(2y²) = 0, y ≥ 1. This defines a smooth even solution for every real x; its derivative at 0 is 0.
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