Class 12 Differential Equations · Short Answer2 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Short Answer #35
Solve y²dx + (x² − xy + y²)dy = 0.
✅ Step-by-Step Solution
Final Answer: arctan(x/y) + ln|y| = C for y ≠ 0; y = 0 is also an integral curve in the original differential form.
- Treat x as dependent on y; put x = vy, so dx/dy = v + y dv/dy.
- The equation becomes v + y dv/dy = −v² + v − 1, hence dv/(1 + v²) = −dy/y. Integration gives arctan v = −ln|y| + C.
- Thus arctan(x/y) + ln|y| = C, y ≠ 0. Substitution in the original differential form also gives the excluded integral curve y = 0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Treat x as dependent on y; put x = vy, so dx/dy = v + y dv/dy.
- [1 mark] The equation becomes v + y dv/dy = −v² + v − 1, hence dv/(1 + v²) = −dy/y. Integration gives arctan v = −ln|y| + C.
- [0.5 marks] Thus arctan(x/y) + ln|y| = C, y ≠ 0. Substitution in the original differential form also gives the excluded integral curve y = 0.
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