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For the differential equation y = y/x + x/y, if its general solution is written as y = x / log|Cx|, then the function φ(x/y) is

1. −x²/y²
2. y²/x²
3. x²/y²
4. −y²/x²

Correct answer: −x²/y²

Solution

From y=x/log|Cx|, with v=y/x=1/log|Cx|, one gets dy/dx - y/x = -1/log^2|Cx| = -v^2, so the equation is dy/dx=y/x+phi(y/x) with phi(t)=-t^2. Evaluating at x/y gives phi(x/y)=-x^2/y^2.

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