CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/2 · Q345 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/2 · Question 34 (Differential Equations)
Solve (x+y)dy+(x−y)dx=0 with y(1)=1.
✅ Step-by-Step Solution
Final Answer: arctan(y/x)+(1/2)ln(x²+y²)=π/4+(1/2)ln 2 on the branch near (1,1).
- Set y=xv for x>0 near 1. Substitution yields x(1+v)v′=−(1+v²).
- Integrate (1+v)dv/(1+v²)=−dx/x to get arctan v+(1/2)ln(1+v²)=−ln x+C.
- Substitute v=y/x and use (1,1) to get C=π/4+(1/2)ln 2.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Set y=xv for x>0 near 1. Substitution yields x(1+v)v′=−(1+v²).
- [3 marks] Integrate (1+v)dv/(1+v²)=−dx/x to get arctan v+(1/2)ln(1+v²)=−ln x+C.
- [1 mark] Substitute v=y/x and use (1,1) to get C=π/4+(1/2)ln 2.
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