Solve Differential equation with absolute value

Question

Hi guys I'm kinda stuck with this problem:

$\frac{dx}{dt} = \sqrt{|x|}$

$x(0)=0$

I could easily solve this if I knew x would be positive but I can't combine solutions for x negative and positive because this is not a linear differential equation right?

I found out that x=constant solves the problem but that doesn't help much.

Answer 1

Hint: Try to look at the cases $x>0$ and $x<0$ separately. The case $x(t)=0$ is a trivial solution.

$$x>0: \dfrac{dx}{dt}=\sqrt{x}$$ $$x<0: \dfrac{dx}{dt}=\sqrt{-x}$$

Answer 2

For positive $x$,

$$\frac{dx}{\sqrt x}=dt$$ and

$$2\sqrt x=t,$$ already taking into account the initial condition.

For negative $x$,

$$\frac{dx}{\sqrt{-x}}=dt$$ and

$$-2\sqrt{-x}=t.$$

You can summarize the solution as

$$x=\frac{\text{sgn}(t)\,t^2}4=\frac{t\,|t|}4.$$