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

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

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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.