What can be said about the uniqueness of an ODE solved with separation of variables

Question

consider the following ODE which can be solved with separation of variables:

$$ x' = 2t (1+x^2), x(0) = 0 $$

The solution is:

$$ \lambda(t) = \tan(t^2) $$ But what can I say about the uniqueness of the solution?

Thank you

Answer 1

The solution is unique, by the Picard–Lindelöf theorem.

Answer 2

The solution is unique because $1+x^2\neq 0$ for any $x$. Hence the variables can be separated and the equation can be integrated.

In general, the uniqueness of the autonomous equations $\dot x=f(x)$ gets broken if for $\hat{x}$ such that $f(\hat{x})=0$ the improper integral $$ \int_{x_0}^\hat{x}\frac{dx}{f(x)} $$ converges.