Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations

Class 12 Maths · Differential Equations · Long Answer #52

Solve (xy − x²)dy = y²dx.

✅ Step-by-Step Solution

Final Answer: ln|y| − y/x = C for xy ≠ 0; additionally y = 0 and the vertical integral curve x = 0.

Treat x as dependent on y for y ≠ 0. Put x = vy, so v + yv′ = v − v².
For v ≠ 0, dv/v² = −dy/y. Integration gives −1/v = −ln|y| + C.
Substitute v = x/y to obtain ln|y| − y/x = C. The division by v omitted x = 0; the division by y omitted y = 0.
Thus the non-axis integral curves have ln|y| − y/x = C; both axes satisfy the original differential form. For graphs y(x), use regular branches where the implicit relation has a finite derivative.

[0.5 marks] Treat x as dependent on y for y ≠ 0. Put x = vy, so v + yv′ = v − v².
[1 mark] For v ≠ 0, dv/v² = −dy/y. Integration gives −1/v = −ln|y| + C.
[1 mark] Substitute v = x/y to obtain ln|y| − y/x = C. The division by v omitted x = 0; the division by y omitted y = 0.
[0.5 marks] Thus the non-axis integral curves have ln|y| − y/x = C; both axes satisfy the original differential form. For graphs y(x), use regular branches where the implicit relation has a finite derivative.

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