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If a curve y = f(x) passes through the point (1, −1) and satisfies the differential equation, y(1 + xy) dx = x dy, then f(−1/2) is equal to:

1. 2/5
2. 4/5
3. −2/5
4. −4/5

Correct answer: 4/5

Solution

Rewriting as dy/dx-y/x=y^2 and substituting v=1/y gives v'+v/x=-1, so xv=-x^2/2+C. Using (1,-1): C=-1/2, giving y=-2x/(x^2+1). Then f(-1/2)=4/5.

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