Differential equation with no unique solution

Question

Can someone help me find solutions for the following differential equation: $x'=-t\,\text{sign}(x)\sqrt{|x|}$ with $x(\tau)=\xi$.

With $x$ a function of $t$ and $\tau$ and $\xi$ constants.

Answer 1

It is possible to compute solutions where $x(t)$ is not changing its sign. Since you wanted to find 'solutions' and not 'all solutions' I will not investigate the case where $x(t)$ changes its sign. (It seems strenuous.)

Now let's take the IVP $$\begin{cases} x'&=-t\,\text{sign}(x)\sqrt{|x|} \\ x(\tau)&=\xi \end{cases}$$

Assuming $x(t)>0$ for all $t$:

We get $x'=-t \sqrt{x}$ which has the solution (by separation of variables) $$x(t)=\frac{1}{16}(t^2-A)^2>0$$ with $x'(t)=\frac{1}{4}(t^2-A)t$ where $A>0$ is a constant that has the properties:

• $x'(t)<0$ i.e. $\begin{cases} \text{for $t>0$: }~t^2-A<0 \Rightarrow t^20 \Rightarrow t^2>A \Rightarrow |t|>\sqrt{A} \Rightarrow t<-\sqrt{A} \end{cases}$
• $x(\tau)=\xi$ i.e. $\begin{cases} \text{for $\tau>0$: }~\tau^2-A=- 4 \sqrt{\xi}<0 \Rightarrow A=\tau^2 + 4\sqrt{\xi} \\ \text{for $\tau<0$: }~\tau^2-A= 4 \sqrt{\xi}>0 \Rightarrow A=\tau^2 - 4\sqrt{\xi} \\ \text{for $\tau=0$: }~ 0-A=-4\sqrt{\xi}<0 \Rightarrow A=4\sqrt{\xi} \end{cases}$

The first condition is problematic for arbitrarily large $t$. Since $A$ is specified by the second condition we get by the first condition that $x(t)=\frac{1}{16}(t^2-A)^2$ is only a solution for $$t \in \begin{cases} \left(-\infty,-\sqrt{\tau^2+4 \sqrt{\xi}} \right) \cup \left(0,\sqrt{\tau^2+4 \sqrt{\xi}} \right) &\text{ if } \tau>0\\ \left(-\infty,-\sqrt{\tau^2-4 \sqrt{\xi}} \right) \cup \left(0,\sqrt{\tau^2-4 \sqrt{\xi}} \right) &\text{ if } \tau<0 \wedge \tau^2>4 \sqrt{\xi} \\ \left(-\infty,-\sqrt{4 \sqrt{\xi}} \right) \cup \left(0,\sqrt{4 \sqrt{\xi}} \right) &\text{ if } \tau=0 \end{cases}$$

Assuming $x(t)<0$ for all $t$:

We get $x'=t\sqrt{-x}$ i.e. $y'=-t\sqrt{y}$ for $y=-x$. So we are back in our first case just with a minus and the solution has to stay negative.

Answer 2

If $t\mapsto x(t)>0$ is a solution defined in some interval $J$ then $$\left(2\sqrt{x(t)}+{1\over2}t^2\right)'={x'(t)\over\sqrt{x(t)}}+t=0\ ,$$ hence $$2\sqrt{x(t)}+{1\over2}t^2={1\over2}c^2$$ for some $c>0$. This allows to conclude that $$\sqrt{x(t)}={1\over4}(c^2-t^2)\qquad(-c

The function $$x(t):\equiv0\tag{2}$$ is a solution as well, but a special one: The Lipschitz-assumption of the general existence and uniqueness theorem is not fulfilled in the points $(\tau,0)$ with $\tau\ne0$. Therefore it does not come as a surprise that IVPs with such a point as initial point have three solution germs starting there: the solution $(2)$ and two arcs of type $(1)$ going off towards $t=0$. In direction $|t|\to\infty$ there is just the trivial behavior $(2)$. See the following figure:

Answer 3

When $x>0$,

$$\frac{x'}{\sqrt x}=-t$$ and

$$x(t)=\left(C-\frac{t^2}4\right)^2=\left(\sqrt{\xi}+\frac{\tau^2-t^2}4\right)^2.$$

When $x<0$,

$$-\frac{x'}{\sqrt {-x}}=-t$$ and

$$x(t)=-\left(C-\frac{t^2}4\right)^2=-\left(\sqrt{-\xi}+\frac{\tau^2-t^2}4\right)^2.$$

And when $x=0$, $x(t)=0$, which is an absorbing state. The solution cannot change sign.