# DE problem: Dog chasing a rabbit

by camilus
Tags: chasing, rabbit
 P: 4 Could someone also derive the equation in relation to y? I'm having huge problems with a question of the sort. Hopefully with the steps laid out it would become clear. Thanks!
 P: 22 So how would I do this if the rabbit has 1/2 the velocity? how would i set up an equation for how X, and Y will change?
Math
Emeritus
 P: 1 Here's another approach that doesn't assume that the dog and rabbit run at constant speeds (but their speeds are the same). Suppose that after some time t>0 the dog has arrived at the point (x,y) and the rabbit has arrived at the point (0,r). Then the distance the dog has travelled is given by $\int^{L}_{x}$$\sqrt{1+(dy/du)^2}du$ and the distance the rabbit has travelled is r. Since they are traveling at the same speed we then have that r=$\int^{L}_{x}$$\sqrt{1+(dy/du)^2}du$. Now, since the dog is heading straight towards the rabbit, we have that dy/dx = (y-r)/x, and so r=y-x(dy/dx), implying that y-x(dy/dx)=$\int^{L}_{x}$$\sqrt{1+(dy/du)^2}du$. Differentiating both sides with respect to x gives the differential equation in part a.