Class 12 Differential Equations · Very Long Answer5 Marksfirst_order_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #108
Solve [y − x cos(y/x)]dy + [y cos(y/x) − 2x sin(y/x)]dx = 0.
✅ Step-by-Step Solution
Final Answer: y² − 2x²sin(y/x) = C, x ≠ 0.
- Let F(x, y) = y² − 2x²sin(y/x), with x ≠ 0.
- Compute F_y = 2y − 2x cos(y/x).
- Compute F_x = −4x sin(y/x) + 2y cos(y/x).
- The given differential form is exactly ½dF, so dF = 0.
- Therefore y² − 2x²sin(y/x) = C. Use regular branches with y − x cos(y/x) ≠ 0 for a finite derivative y(x).
📋 CBSE Stepwise Marking Scheme
- [1 mark] Let F(x, y) = y² − 2x²sin(y/x), with x ≠ 0.
- [1 mark] Compute F_y = 2y − 2x cos(y/x).
- [1 mark] Compute F_x = −4x sin(y/x) + 2y cos(y/x).
- [1 mark] The given differential form is exactly ½dF, so dF = 0.
- [1 mark] Therefore y² − 2x²sin(y/x) = C. Use regular branches with y − x cos(y/x) ≠ 0 for a finite derivative y(x).
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