Consider the ODE $$\dot \gamma(t) = \sqrt{|\gamma(t)|}$$ with initial data $\gamma(0) =x_0 = -c^2$.
Having fixed one value of $c$, the solutions of this Cauchy problem are not unique because one has $x(t) = -(t/2 - c)^2$ for $0 \le t \le 2c$, then the solution can stay at the origin for some time $2T(c)$ (also maybe infinite or zero) and then continue as $x(t) = (t/2 - T (c) - c)^2$ for $t \ge 2c + 2T(c)$.
- How can I draw these multiple characteristic lines of this problem by using Matlab? And how can I highlight (maybe in a different color) some of them? For example the one with $T(c) = 0$ and the one with $T(c) = \infty$?
- Usually, what numerical methods are used to solve (more complicated) ODEs where non-uniqueness of solutions occurs?