Initial statement is:
Infinitely many functions are solutions of x'=x^(2/3), all with x(0)=0.
We construct the following:
(it works for every c>0, so that's an infinite amount):
For x < c, take x=0 as solution, for x>=c take x=(t-c)³/27.
Now the following statements are true, for every c >0:
1. x(0)=0 (we defined it that way)
2. From t = -infty up until t=c, x'=0, which is equal to x^(2/3) (constant solution)
3. If t>=c, x=(t-c)³/27. Now also x'=x^(2/3) (we checked it)
4. for t=c, x'=0, so there is a smooth transition
Conclusion: we've found an infinite amount of functions, defined for all real numbers, that satisfy the diff. eq. and the extra condition x(0)=0.
Looks like a proof to me.