Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #49
Solve xe^(y/x) − y + xy′ = 0.
✅ Step-by-Step Solution
Final Answer: e^(−y/x) = ln|x| + C > 0, x ≠ 0.
- Put v = y/x, so y′ = v + xv′.
- The equation reduces to xv′ = −eᵛ. Separate e^(−v)dv = −dx/x.
- Integrate −e^(−v) = −ln|x| + C₁.
- Thus e^(−y/x) = ln|x| + C > 0; equivalently y = −x ln(ln|x| + C).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put v = y/x, so y′ = v + xv′.
- [1 mark] The equation reduces to xv′ = −eᵛ. Separate e^(−v)dv = −dx/x.
- [1 mark] Integrate −e^(−v) = −ln|x| + C₁.
- [0.5 marks] Thus e^(−y/x) = ln|x| + C > 0; equivalently y = −x ln(ln|x| + C).
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