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A curve passes through (1, -2) and at every point (x, y) its tangent has slope (x² - 2y)/x. Through which of the following points does the curve also pass?

1. (-sqrt(2), 1)
2. (sqrt(3), 0)
3. (-1, 2)
4. (3, 0)

Correct answer: (sqrt(3), 0)

Solution

Solving the linear ODE gives x² y = x⁴/4 + C; using (1, -2) gives C = -9/4, so y = x²/4 - 9/(4x²), which passes through (sqrt(3), 0).

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