Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations

Class 12 Maths · Differential Equations · Long Answer #65

Show that (x − y)y′ = x + 2y is homogeneous and solve it.

✅ Step-by-Step Solution

Final Answer: −½ln(x² + xy + y²) + √3 arctan((2y + x)/(√3x)) = C locally for x ≠ 0, x ≠ y.

The ratio (x + 2y)/(x − y) depends only on v = y/x, hence is homogeneous. Put y = xv.
Then xv′ = (1 + v + v²)/(1 − v), so (1 − v)dv/(v² + v + 1) = dx/x.
Write 1 − v = −½(2v + 1) + 3/2. Integration gives −½ln(v² + v + 1) + √3 arctan((2v + 1)/√3) = ln|x| + C.
Substitution gives −½ln(x² + xy + y²) + √3 arctan((2y + x)/(√3x)) = C. This is a local chart; use continuous angle branches across x = 0 and exclude x = y where the original graph equation is singular.

[0.5 marks] The ratio (x + 2y)/(x − y) depends only on v = y/x, hence is homogeneous. Put y = xv.
[1 mark] Then xv′ = (1 + v + v²)/(1 − v), so (1 − v)dv/(v² + v + 1) = dx/x.
[1 mark] Write 1 − v = −½(2v + 1) + 3/2. Integration gives −½ln(v² + v + 1) + √3 arctan((2v + 1)/√3) = ln|x| + C.
[0.5 marks] Substitution gives −½ln(x² + xy + y²) + √3 arctan((2y + x)/(√3x)) = C. This is a local chart; use continuous angle branches across x = 0 and exclude x = y where the original graph equation is singular.

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