Many times in mathematics, as for example when we find the solution of an ODE, we can not claim uniqueness just by construction, instead we have to use a theorem.
The reasoning behind this is that even if we found a solution and the solution appears to be unique from the point of view of the method used, how can we be sure there is no another method which provides another solution?
Now my question: sometimes the following type of argument is accepted as valid. For example, a simple differential equation like this one $y'(x)=x^2$ with $y(0)=0$, we say that $y(x)=\frac{1}{3}x^3$ is the only solution (without the use of an uniqueness theorem, I think is because of the Fundamental Theorem of Calculus).
How can be sure now that there is not another way to solve the equation that provides another solution (without using the uniqueness theorem of course)?
When is this type of argument valid and when not?
Down in some of the answers some people say that the reason in the previous example for uniqueness is because the anti-derivative is unique, which sounds reasonable.
But the Laplace transform of a function that satisfies certain conditions is also unique. And we can't say that the solution of an ODE is unique only beause it was calculated through the Laplace transform method.
It's unique because it is a ODE with separable variables, and its solution is found via separation of the variables and integration. The function obtained via integration is unique except for the arbitrary constant, which is uniquely determined by the initial condition.
EDIT: I must correct this: let's suppose you have an ODE which is variable-separable that is defined by $$ h(y)y'=g(x) $$ You can find a solution via integration as usual: $$ \int h(y)dy=\int g(x)dx $$ The function is given inplicitly by the above equation. Let's suppose that $y(a)=b$ is the initial condition. If $y'(a)\neq 0$ then $y$ is locally invertible (as per the Inverse Function theorem) and you can find a local unique solution $y(x)$ which can be extended to a maximal conected domain; in this case it is correct to say that the solution obtained via the separation method is unique without recurring to the Existence & Uniqueness of solutions.
An important remark: the Existence and Uniqueness Theorem for ODEs works fine for equations defined as $y'(x)=f(y,x)$, where $|f|$ is limited on a neighbourhood of the initial condition $(a,y(a))$. In the example cited on the comments, $f$ is something like $$ f(x,y)=\dfrac{\hat{f}(x)}{y} $$ for some $\hat{f}$ so if $y(0)=0$, $|f|$ is not limited around $(0,0)$ and uniqueness cannot be guaranteed.