For instance, consider the following initial value problem $$x'=3x^{2/3}, \ \ \ \ x(0)=0$$
This initial value problem has infinitely many solutions given by $$ x(t) = \begin{cases}
0 & t<c \\
(t- c)^3 & t \geq c
\end{cases} \ \ \ \ \ (*) $$ for any non-negative $c$. Picard's existence theorem says that the initial value problem $x'=f(x,t), x(t_0)=x_0$ has a unique solution if $f(x,t)$ is continuous at $(x_0,t_0)$ and Lipschitz continuous in $x$. So the non-uniqueness comes from the fact that $3x^{2/3}$ is not Lipschitz at $x=0$. But I was wondering why a student taking a basic undergrad course in differential equations is not able to get the totality of solutions (*) ? i.e. one proceeds like this $$ \frac{1}{3}x^{-2/3}dx=dt \ \ \ \Rightarrow \ \ \ \ x^{1/3}=t+C $$ applying the initial condition we get $C=0$. So $$x(t)=t^3$$
Thanks
Best Answer
In the derivation that the undergraduate student has found (which is sometime called separation of variables) he is assuming that $x\neq 0$. In fact the solution he finds: $$ x(t) = (t+C)^3 $$ is valid and unique in every point where $x(t)\neq 0$. The same undergraduate student, before dividing the equation by $x$, would check that he is not loosing solutions. And he finds, that $$ x(t) = 0 $$ is in fact a solution.
Now he notices that solutions of the first kind can touch the solution $x=0$ and when this happen the derivatives of the two solutions agree. Hence he can merge the solutions together to get the general solution you wrote at the beginning.