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By eliminating the arbitrary constant c, find the differential equation whose general solution is x = c*y*e^(xy).
- dy/dx = y*(1 - xy) / (x*(1 + xy))
- dy/dx = x*(1 - xy) / (y*(1 + xy))
- dy/dx = y*(1 + xy) / (x*(1 - xy))
- dy/dx = (1 - xy) / (1 + xy)
Correct answer: dy/dx = y*(1 - xy) / (x*(1 + xy))
Solution
Taking logs turns x = c*y*e^(xy) into ln(x) = ln(c) + ln(y) + xy. Differentiating eliminates the constant c automatically. The result is rearranged to express dy/dx purely in terms of x and y, giving a first-order differential equation that the original family of curves satisfies.
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