I need to find all solutions of Cauchy problem $\dot {x}=b(x), x(0)=0, b\in C(\mathbb{R}), b(0)=0, b\ne0$ if $x\ne0$.
$\frac{dx}{dt}=b(x)$, so divide by $b(x)\ne0$ we get $\frac{dx}{b(x)}=1*dt$. Then I can integrate both parts. $\int_{x_0}^{x}\frac{dx}{b(x)}$=$t-t_0$, so I get the equation on $x$ and $t$. This solution seems to be too easy to be true. And I'm not sure that I have found all of them.
Obviously, $x(t)=0$ is a solution to the initial value problem. The question remains if it is the only one. If $b$ is differentiable or at least Lipschitz around $x=0$ then the uniqueness theorem can be applied. On the other hand, you have $b(x)=2\sqrt{|x|}$ which has $x(t)=t^2$ as one of the additional solutions. You can try to get stronger results from the type of the singularity in the integrand.