Find the DE for C

Question

A motorboat starts at the origin and moves in the direction of the positive x-axis, pulling a waterskier along a curve C called a tractrix. See the figure below. The waterskier, initially located on the y-axis at the point (0, a), is pulled by a rope of constant length a that is kept taut throughout the motion. At time t > 0 the waterskier is at point P(x, y). Assume that the rope is always tangent to C. Use the concept of slope to determine a differential equation for the path C of motion.

Curve C figure

Answer 1

When the waterskier is located at the point $(x,y)$ the motorboat is located at the point $\Big(x+\sqrt{a^2-y^2},0\Big)$.

The line connecting the waterskier with the motorboat has slope $-\frac{y}{\sqrt{a^2-y^2}}$ which happens to be slope of the tangent line to $C$ at $(x,y)$.

So, to find an expresion for $C$, we'll solve the initial value problem $$\frac{dy}{dx}=-\frac{y}{\sqrt{a^2-y^2}}; y(0)=a$$ This DE is separable and has solution $$a\ln \Bigg(\frac{y}{a+\sqrt{a^2-y^2}}\Bigg)+\sqrt{a^2-y^2}=-x$$ Click here to see this.