Phase Plane — Question 2

Question 2 – Solution

Strategy. Keep zero-coordinate cases before separating variables, and use explicit signs to orient each connected branch.

Step 1: Find a division-free invariant. The product rule gives $(xy)'=x'y+xy'=xy-xy=0$. Thus $xy=C$ along every solution, including axis solutions. The only equilibrium is $(0,0)$.

Step 2: Recover the axis trajectories. If $y=0$, then $x=x_0e^t$; each nonzero horizontal half-axis is a trajectory directed away from the origin. If $x=0$, then $y=y_0e^{-t}$; each nonzero vertical half-axis is directed toward the origin. The origin itself is a constant trajectory. Division by $x$ loses the vertical axis; subsequent division by $y$ can also lose the horizontal axis.

Step 3: Separate the hyperbola branches. For nonzero coordinates, $x=x_0e^t$ and $y=y_0e^{-t}$ preserve both signs. In quadrants I, II, III, IV the directions are respectively right/down, left/down, left/up, right/up. Each fixed nonzero $C$ has two disconnected branches; no continuous solution switches between them. The equation $xy=0$ likewise combines several distinct trajectories rather than describing one orbit through the origin.

Step 4: Apply the initial data and characterize attraction. For the stated IVP, $$\boxed{x=2e^t,\qquad y=\tfrac 12e^{-t},\qquad xy=1.}$$ As $t\to\infty$, $x\to\infty$, $y\to 0$; as $t\to-\infty$, $x\to 0$, $y\to\infty$. Neither limit is the origin. For general initial data, convergence to the origin in forward time occurs exactly when $\boxed{x_0=0}$, including the zero state. The figure distinguishes the axis directions from representative hyperbola branches.