Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #103
Solve y′ = (x + y)/(x − y).
✅ Step-by-Step Solution
Final Answer: arctan(y/x) − ½ln(x² + y²) = C locally, excluding x = y.
- For x ≠ 0 put v = y/x. Then v + xv′ = (1 + v)/(1 − v).
- Hence (1 − v)dv/(1 + v²) = dx/x.
- Integrate arctan v − ½ln(1 + v²) = ln|x| + C.
- Thus arctan(y/x) − ½ln(x² + y²) = C on a local angle chart; continue the angle continuously across x = 0 and exclude x = y.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For x ≠ 0 put v = y/x. Then v + xv′ = (1 + v)/(1 − v).
- [1 mark] Hence (1 − v)dv/(1 + v²) = dx/x.
- [1 mark] Integrate arctan v − ½ln(1 + v²) = ln|x| + C.
- [0.5 marks] Thus arctan(y/x) − ½ln(x² + y²) = C on a local angle chart; continue the angle continuously across x = 0 and exclude x = y.
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