CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1 · Q345 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1 · Question 34 (Differential Equations)
Solve \(x^2dy+y(x+y)dx=0\), with \(y(1)=1\).
✅ Step-by-Step Solution
Final Answer: y=2x/(3x²−1) near x=1.
- Set y=xv for x>0 near 1. Then xv′=−v(v+2).
- Separate dv/[v(v+2)]=−dx/x and use 1/[v(v+2)]=(1/2)(1/v−1/(v+2)). Thus ln abs(v/(v+2))=−2 ln x+C. The initial value v(1)=1 gives v/(v+2)=1/(3x²).
- Solve v=2/(3x²−1) and multiply by x. The interval through 1 is (1/√3,∞).
📋 CBSE Stepwise Marking Scheme
- [1 mark] Set y=xv for x>0 near 1. Then xv′=−v(v+2).
- [3 marks] Separate dv/[v(v+2)]=−dx/x and use 1/[v(v+2)]=(1/2)(1/v−1/(v+2)). Thus ln abs(v/(v+2))=−2 ln x+C. The initial value v(1)=1 gives v/(v+2)=1/(3x²).
- [1 mark] Solve v=2/(3x²−1) and multiply by x. The interval through 1 is (1/√3,∞).
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