Exam-Style Problem
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Nov 2016 p33 q5
2296
The diagram shows a variable point \(P\) with coordinates \((x, y)\) and the point \(N\) which is the foot of the perpendicular from \(P\) to the \(x\)-axis. \(P\) moves on a curve such that, for all \(x \geq 0\), the gradient of the curve is equal in value to the area of the triangle \(OPN\), where \(O\) is the origin.
(i) State a differential equation satisfied by \(x\) and \(y\).
The point with coordinates \((0, 2)\) lies on the curve.
(ii) Solve the differential equation to obtain the equation of the curve, expressing \(y\) in terms of \(x\).
(iii) Sketch the curve.
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