My try on this initial value problem: $\dot{x}=\sqrt{|{x}|}$, $x(0)=0$.

Question

I'd like you to check if my try on this problem is correct. Here goes the statement:

Given the initial value problem $\dot{x}=\sqrt{|{x}|}$, $x(0)=0$:

$a)$ Give a solution to this IVP.

$b)$ Is unique?

$c)$ In case of a negative answer on $b)$, does this contradicts Picard's theorem of existence and uniqueness?

Ok, so here is my approach:

$a)$ We see that the critical point is on $(0,0)$. I see two solutions to this problem:

\begin{cases}-t^2/4\ \ \ \ if\ \ \ t<0 \\ t^2/4\ \ \ \ if\ \ \ t\geq0\end{cases}

or

\begin{cases}-t^2/4\ \ \ \ if\ \ \ t\leq0 \\ t^2/4\ \ \ \ if\ \ \ t>0.\end{cases}

$b)$ Is unique? No, we have a problem on $(0,0)$, as I stated before.

$c)$ Does this contradicts Picard's theorem? No, because $\sqrt{x}$ is not Lipschitz on $t=0$.

Is that correct?

Thanks for your time.

Answer 1

Both functions are identical as they have the same values everywhere, including at $t=0$. You get the additional solutions (with $a,b>0$, including one or both $\infty$) $$ y(t)= \begin{cases} -\frac{(t+a)^2}4& \text{ for }&t\le -a \\ 0&\text{ for }&-a < t<b \\ \frac{(t-b)^2}4 &\text{ for }&b \le t \end{cases} $$