Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #44
Solve xy′ = y(ln y − ln x + 1).
✅ Step-by-Step Solution
Final Answer: y = xe^(Cx), x > 0, C ∈ ℝ.
- The original logarithms require x > 0 and y > 0. Put v = y/x > 0, so y′ = v + xv′.
- Substitution yields xv′ = v ln v. Set u = ln v, giving u′ = u/x, without excluding u = 0.
- Solve (u/x)′ = 0 to obtain u = Cx, with every real C including zero.
- Therefore y = xe^(Cx), x > 0. This includes the previously lost solution y = x.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The original logarithms require x > 0 and y > 0. Put v = y/x > 0, so y′ = v + xv′.
- [1 mark] Substitution yields xv′ = v ln v. Set u = ln v, giving u′ = u/x, without excluding u = 0.
- [1 mark] Solve (u/x)′ = 0 to obtain u = Cx, with every real C including zero.
- [0.5 marks] Therefore y = xe^(Cx), x > 0. This includes the previously lost solution y = x.
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