Systems of Differential Equations — Question 8

Question 8 – Solution

Strategy. Prove nonuniqueness by constructing different differentiable solutions; do not infer it solely from a theorem’s failure.

Step 1: Test the regularity hypothesis. The field $F(x,y)=(2\sqrt{|x|},-y)^T$ is continuous everywhere. For $h>0$, the change in its first component between $(h,y)$ and $(0,y)$, divided by the state distance, is $2\sqrt h/h=2/\sqrt h$. This is unbounded as $h\downarrow 0$, so no local Lipschitz constant exists near $x=0$. The quoted uniqueness theorem cannot guarantee uniqueness there.

Step 2: Construct and check the waiting family. For each $a\ge 0$, define $$\boxed{x_a(t)=\begin{cases}0,&0\le t\le a,\\(t-a)^2,&t>a.\end{cases}}$$ Before $a$, both $x_a'$ and $2\sqrt{|x_a|}$ are zero. After $a$, both are $2(t-a)$. At $a$, the left and right derivatives are zero, so the function is continuously differentiable and satisfies the equation there too. At $a=0$, the same check uses the right derivative at the endpoint.

Step 3: Check the common data. The second scalar equation uniquely gives $\boxed{y(t)=e^{-t}}$. Every pair $(x_a,e^{-t})$ has initial state $(0,1)$ and initial derivative $(0,-1)$. The pair $(0,e^{-t})$ is another solution, corresponding to waiting forever. Different finite waiting times produce different first components, despite identical initial states and slopes.

Step 4: Test whether one extra observation selects a solution. Every $a\ge 1$ gives $x_a(1)=0$, as does the never-departing solution. For example, $a=1$ and $a=2$ satisfy all the data, but at $t=3/2$ their first components are $1/4$ and $0$. Uniqueness is still absent. A complete state specifies all initial coordinates; unique evolution also requires appropriate properties of the differential equations. The figure shows three members of this family, with time increasing rightward.