Is there a solution for the following initial value problem?

Hello :)

I don't know how to solve this one:

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Is this statement correct or wrong? Prove or try to reason your answer.

Let $\displaystyle f(x): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be a bounded $\displaystyle C^1$ vector field. Then there exists a solution of the initial value problem

$\displaystyle x'=f(x), x(0)=x_0$

for all $\displaystyle t \ge 0$.

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My guess is that this statement is false. It is not the fundamental theorem of ordinary differential equations as we had it in lectures because f(x) is not linear. If f(x) was linear that would be exactly this fundamental theorem of ordinary differential equations.

Is my guess correct?

I tried a couple of functions which are continuous differentable but didn't find one where I can prove that the solution doesn't exists. For example:

$\displaystyle f(x_1,x_2):=(log(x_1),x_2): \mathbb{R}^+ \times \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R^+}$

I don't know how to calculate x(t) for this ivp but mathematica says there is a solution.

What do you think of this? Is the statement true?

Regards