allerySolve dy/dx = (x + y)/(x - y) (homogeneous, deg 0):
AImplicit: ln(x² + y²) + 2 arctan(y/x) = C
Bx² + y² = C
Cy = x
Dy = -x
Answer & Solution
Correct answer: A. Implicit: ln(x² + y²) + 2 arctan(y/x) = C
Let v = y/x.
We have dy/dx = v + x dv/dx.
So dy/dx = (1 + v)/(1 - v).
**Solve for dv:** dv/dx = [(1+v) - v(1-v)] / [x(1-v)] = (1 + v²)/[x(1-v)].
**Separate:** (1-v)/(1+v²) dv = dx/x.
**Integrate:** arctan v - (1/2) ln(1+v²) = ln x + C. Back-substitute v = y/x: arctan(y/x) - (1/2) ln(1 + y²/x²) = ln x + C
→ equivalent form ln(x²+y²) - 2 arctan(y/x) = C'.
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