The applet below shows a brown boat in the water. A blue rope of fixed length (that's always held taught) connects the brown boat to a red point. This red point stays on the gray dock and is allowed to move vertically along the dock.
Go ahead and drag the red point upwards and observe the path traced out by the the brown boat.
Feel free to do this for different values of taught rope length (adjustable by the blue slider.)
After interacting with this applet for a few minutes, please answer the questions that appear below it.

1) The tractrix is also called a curve of pursuit.
Can you explain, from your observations, why this label makes sense?
2) Suppose point P (the boat) has coordinates (x, y).
Suppose the red point A has coordinates (0,a).
Write an expression for the slope of the blue rope (segment AP) in the applet above.
3) Solve the differential equation you wrote for (2) above.