I need to show that the initial value problem:
$\dot{x}=|x|^{1/2}$
$x(0)=0$
has 4 different solutions through the point (0,0). The problem also says that I have to sketch the solutions in the $(t,x)$-plane.
My solution:
So basically I have no issue with solving the problem, I just want to make sure I am not missing 2 extra solutions, or if I am just misinterpreting the problem.
After solving the first equation with elemental calculus, I get: $x(t)=\pm t^2/4$. But these are just 2 curves. However when I do the plot, the point (0,0) is arrived by 4 different curves: $\pm t^2/4$, split in half, and with other two cases of $t>0$ and $t<0$. So is my solution correct, or am I missing something?
The initial value problem $$ x'=|x|^{1/2}, \quad x(0)=0, $$ does not have $4$ but infinite many solutions.
Nevertheless, the $4$ ones you are referring to are: $$ \psi_1(t)=0,\quad \psi_2(t)=\frac{t^2}{4}\mathrm{sgn}\,t, \quad \psi_3=\max\{\psi_1,\psi_2\} \quad\text{and}\quad \psi_4=\min\{\psi_1,\psi_2\}. $$