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Which equation describes a curve such that the segment intercepted on the y-axis by any tangent is proportional to the square of the y-coordinate of the point where the tangent touches the curve?
- x + y = cxy
- 1/x + 1/y = c
- A/x + B/x = xy
- A/x + B/y = 1
Correct answer: A/x + B/y = 1
Solution
The tangent's y-intercept is Y=y-x*y'; setting Y=k*y^2 gives y - x y' = k y^2. Solving this ODE yields 1/y = A/x + B (constants), which rearranges to the form A/x + B/y = 1.
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