Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #69
Solve y′ = −(x + y cos x)/(1 + sin x), y(0) = 1.
✅ Step-by-Step Solution
Final Answer: y = (2 − x²)/[2(1 + sin x)], −π/2 < x < 3π/2.
- Write y′ + [cos x/(1 + sin x)]y = −x/(1 + sin x), with 1 + sin x ≠ 0.
- IF = 1 + sin x, so [(1 + sin x)y]′ = −x.
- Integrating gives (1 + sin x)y = −x²/2 + C. At (0, 1), C = 1.
- Thus y = (2 − x²)/[2(1 + sin x)] on (−π/2, 3π/2), the maximal interval containing 0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Write y′ + [cos x/(1 + sin x)]y = −x/(1 + sin x), with 1 + sin x ≠ 0.
- [1 mark] IF = 1 + sin x, so [(1 + sin x)y]′ = −x.
- [1 mark] Integrating gives (1 + sin x)y = −x²/2 + C. At (0, 1), C = 1.
- [0.5 marks] Thus y = (2 − x²)/[2(1 + sin x)] on (−π/2, 3π/2), the maximal interval containing 0.
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