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Consider an ODE of the form $$y''=f(x,y,y') $$ with $x \geq 0$, and initial conditions of the form $$y(0)=y_0>0, \\y'(0)=m_0. $$ I want to claim that under reasonable conditions (which I can't precisely formulate at the moment) on the function $f$, we have the following

The solution $y$ with the initial conditions above will attain a negative value in its interval of existence, provided that $m_0$ is sufficiently large and negative.

I was wondering if such an existence result for boundary value problems is known. If so, I'd like to see a reference. Otherwise, I'd like help in formulating the statement properly.

Thanks in any case!

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