Chapter 9.P, Problem 9P

A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system (as shown in the figure), assume:

(i) The rabbit is at the origin and the dog is at the point (L, 0) at the instant the dog first sees the rabbit.

(ii) The rabbit runs up the y-axis and the dog always runs straight for the rabbit.

(iii) The dog runs at the same speed as the rabbit.

(a) Show that the dog’s path is the graph of the function $y=f\left(x\right)$, where y satisfies the differential equation

$x\frac{d^{2}y}{d{x}^2}=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

(b) Determine the solution of the equation in part (a) that satisfies the initial conditions $y=y^{\prime }=0$ when $x=L$. [Hint: Let $z=dy/dx$ in the differential equation and solve the resulting first-order equation to find z; then integrate z to find y.]

(c) Does the dog ever catch the rabbit?