Class 12 Differential Equations · Very Long Answer5 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #115
Solve (y − sin x)dx + tan x dy = 0, y(0) = 0.
✅ Step-by-Step Solution
Final Answer: y = ½sin x, −π/2 < x < π/2.
- For sin x ≠ 0 and cos x ≠ 0, write y′ + y cot x = cos x.
- Use IF sin x, giving (y sin x)′ = sin x cos x.
- Integrate y sin x = ½sin²x + C.
- A finite continuous value y(0) = 0 forces C = 0. Hence y = ½sin x for x ≠ 0 near the initial point.
- The same formula extends differentiably through x = 0 and directly satisfies the original undivided equation. On (−π/2, π/2), y = ½sin x.
📋 CBSE Stepwise Marking Scheme
- [1 mark] For sin x ≠ 0 and cos x ≠ 0, write y′ + y cot x = cos x.
- [1 mark] Use IF sin x, giving (y sin x)′ = sin x cos x.
- [1 mark] Integrate y sin x = ½sin²x + C.
- [1 mark] A finite continuous value y(0) = 0 forces C = 0. Hence y = ½sin x for x ≠ 0 near the initial point.
- [1 mark] The same formula extends differentiably through x = 0 and directly satisfies the original undivided equation. On (−π/2, π/2), y = ½sin x.
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