How exactly do you solve the ODE : $\dot x = -\sqrt{|x|}$ How do you deal with $ \dot x = -\sqrt{|x|}$
The square root is irksome and the absolute value inside raises serious eyebrows. The function is continuous but has sort of a nondifferentiable point at zero. No one knows how to visualize this function.
Proceeding as usual is a deadend:
$\dfrac{dx}{\sqrt{|x|}} = -dt$
What to do?
 A: For $x>0$, 
\begin{align*}
  \frac{dx}{dt} &=-\sqrt{x} \\
  \frac{dx}{\sqrt{x}} &=-dt \\
  2\sqrt{x} &= a-t  \\
  x &= \left( \frac{a-t}{2} \right)^{2} \:,\quad t<a
\end{align*}
For $x<0$, 
\begin{align*}
  \frac{dx}{dt} &=-\sqrt{-x} \\
  \frac{dx}{\sqrt{-x}} &=-dt \\
  -2\sqrt{-x} &= b-t \\
  2\sqrt{-x} &= t-b \\
  x &= -\left( \frac{t-b}{2} \right)^{2} \:,\quad t>b
\end{align*}
In general the solution space can be connected ($a=b$) or disconected semi-parabolae ($a\neq b$) subject to different initial/boundary conditions.
Edit : 
Thanks to all the commenters, continuity can be achieved by setting $x(t)=0, \forall t\in [a,b]$ provided $a\leq b$. In case of $a=-1,b=2$, we have

A: use the separation of variable to get the solution. all solution are decreasing. solution starting at a positive value at time $0$ eventually becomes zero in finite time.
the differential equation $$\dot x = -\sqrt{|x|}, x(0) = x_0$$ has the solution 
$$x = \left\{ \begin{array}{ll} 
\left(\sqrt{x_0} - \frac t2\right)^2 & \text{ if $x_0 > 0, t \le 2\sqrt x_0$ }\\
0 &\text{ if $x_0 > 0$ } \\
 -\left(\sqrt{-x_0}- \frac t2\right)^2 & \text{ if $x_0 < 0, t \ge -2\sqrt{-x_0}$ }
\end{array}
\right.$$
