A rabbit is hiding at $(0,1)$ and a fox is standing at the point $(0,-1)$ and a big tree is at $(0,0)$ so that rabbit is safe from the attack of Fox.
Now the Fox moves in horizontal direction at a speed of $1$ unit along the line $y=-1$ towards negative infinity from $(0,-1)$ .
To avoid Detection, rabbit will run towards its right but in a curved path at a constant speed of $2$ units so as to keep Tree between itself and Fox.
Form the Differential equation of the path of Rabbit.
My try:
Let any point on the path of rabbit be $P(r \cos \theta, r \sin \theta)$
Also any point on path of Fox is $Q(a,-1)$
according to my thinking the points $P$,$O$ and $Q$ are always collinear.
So
$$\begin{vmatrix} 0 & 0 &1 \\ a & - 1&1 \\ r \cos \theta&r \sin \theta & 1 \end{vmatrix}=0$$
which gives $$a=-\cot \theta$$
So the point $Q$ is $Q(-\cot \theta, -1)$
any clue from here?
The movement is described by
$$ \dot x^2+\dot y^2 = v_r^2\\ \frac{x}{y} = \frac{-v_f t}{-1} $$
here $v_r = 2,\ \ v_f = 1$
Attached the rabbit evasion curve in green
In dashed red the opposition at $t = 0.5$
The MATHEMATICA script
parms = {vf -> 1, vr -> 2}; equs = {x'[t]^2 + y'[t]^2 == vr^2, x'[t] == D[y[t] vf t, t]} /. parms tmax = 1; soly = NDSolve[{equs, x[0] == 0, y[0] == 1}, {x, y}, {t, 0, tmax}][[2]]; gr1 = ParametricPlot[Evaluate[{x[t], y[t]} /. soly], {t, 0, tmax}, PlotStyle -> {Green, Thick}]; gr2 = ParametricPlot[{-t, -1}, {t, 0, tmax}, PlotStyle -> {Blue, Thick}]; p1 = Evaluate[{x[t], y[t]} /. soly] /. {t -> 0.5}; p2 = {-1 t, -1} /. {t -> 0.5}; p0 = {0, 0}; grp1 = Graphics[{Red, Disk[p1, 0.02]}]; grp2 = Graphics[{Red, Disk[p2, 0.02]}]; grp0 = Graphics[{Red, Disk[p0, 0.02]}]; grp3 = ListLinePlot[{p1, p2}, PlotStyle -> {Red, Dashed}]; Show[gr1, gr2, grp1, grp2, grp3, grp0, PlotRange -> All, AxesOrigin -> {0, 0}]