Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #47
Solve (1 + x²)y′ + 2xy − 4x² = 0, y(0) = 0.
✅ Step-by-Step Solution
Final Answer: y = 4x³/[3(1 + x²)], x ∈ ℝ.
- In linear form P = 2x/(1 + x²), so IF = 1 + x².
- Multiplication gives [(1 + x²)y]′ = 4x²; integration gives (1 + x²)y = 4x³/3 + C.
- Use y(0) = 0 to obtain C = 0.
- Therefore y = 4x³/[3(1 + x²)] for all real x.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] In linear form P = 2x/(1 + x²), so IF = 1 + x².
- [1 mark] Multiplication gives [(1 + x²)y]′ = 4x²; integration gives (1 + x²)y = 4x³/3 + C.
- [1 mark] Use y(0) = 0 to obtain C = 0.
- [0.5 marks] Therefore y = 4x³/[3(1 + x²)] for all real x.
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