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