Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #46
Solve xy′ = y − x tan(y/x), y(1) = π/4.
✅ Step-by-Step Solution
Final Answer: x sin(y/x) = 1/√2; near x = 1, y = x arcsin(1/(√2x)), x > 1/√2.
- Put v = y/x. Substitution gives xv′ = −tan v.
- Integrate cot v dv = −dx/x: ln|sin v| = −ln|x| + C, hence x sin v = C₁.
- At (1, π/4), C₁ = 1/√2. Select the arcsine branch through π/4.
- The solution is y = x arcsin(1/(√2x)), on x > 1/√2; cos(y/x) must remain nonzero.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put v = y/x. Substitution gives xv′ = −tan v.
- [1 mark] Integrate cot v dv = −dx/x: ln|sin v| = −ln|x| + C, hence x sin v = C₁.
- [1 mark] At (1, π/4), C₁ = 1/√2. Select the arcsine branch through π/4.
- [0.5 marks] The solution is y = x arcsin(1/(√2x)), on x > 1/√2; cos(y/x) must remain nonzero.
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