Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #105
Solve x²y dx − (x³ + y³)dy = 0.
✅ Step-by-Step Solution
Final Answer: ln|y| − x³/(3y³) = C for y ≠ 0; y = 0 is an additional integral curve.
- Treat x as dependent on y ≠ 0 and put u = x³, so u′ = 3x²dx/dy.
- The equation gives u′ − 3u/y = 3y². IF = y^(−3).
- Integrate (u/y³)′ = 3/y to obtain x³/y³ = 3ln|y| + C₁.
- Thus ln|y| − x³/(3y³) = C; also retain y = 0 from the original differential form. For y(x), restrict to regular implicit branches.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Treat x as dependent on y ≠ 0 and put u = x³, so u′ = 3x²dx/dy.
- [1 mark] The equation gives u′ − 3u/y = 3y². IF = y^(−3).
- [1 mark] Integrate (u/y³)′ = 3/y to obtain x³/y³ = 3ln|y| + C₁.
- [0.5 marks] Thus ln|y| − x³/(3y³) = C; also retain y = 0 from the original differential form. For y(x), restrict to regular implicit branches.
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