CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Q303 Markshomogeneous_equations
CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Question 30 (Differential Equations)
Solve (3xy+y²)dx+(x²+xy)dy=0. OR Solve x(1+y²)dx−y(1+x²)dy=0 through (1,0).
Alternative 1 3 Marks
Solve (3xy+y²)dx+(x²+xy)dy=0.
Answer: x²y(2x+y)=C on regular branches with x≠0.
- Multiply the differential form by 2x.
- The result is the total differential of 2x³y+x²y².
- Integrate to get x²y(2x+y)=C. The constant-zero levels include the straight solutions y=0 and y=−2x.
Alternative 2 3 Marks
Solve x(1+y²)dx−y(1+x²)dy=0 through (1,0).
Answer: 1+x²=2(1+y²). The curve has a vertical tangent at (1,0).
- Divide by the positive product (1+x²)(1+y²).
- Integrate to get ln(1+x²)−ln(1+y²)=constant.
- Use (1,0) to get the ratio 2. The curve can locally be written as x(y)=√(1+2y²).