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Given:

$dx/dt=x^2+t^2$

$x(0)=0$

What's the correct way to analyze this problem? I understand that to check if the solution exists I have to check its continuity.

$f(t,x)=x^2+t^2$

$\frac{\partial f}{\partial x} =2x $

This means that this problem has a solution since $f(t,x)$ and $\frac{\partial f}{\partial x}$ are continous everywhere.

Is this the correct way to analyze this kind of problems or do I also have to check the interval where the initial condition exists?

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    you must look for the Theorem of Picard-Lindeloef – 2017-01-30
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    You have it right. Check for regions where $f$ and $df/dy$ exist, your solution exists in the region in which the initial condition is. – 2017-01-30
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    Thanks @Kaynex, what I don't understand is how do I check for these regions? Do I have to evaluate $f$ and $f'$ with x(0) ? – 2017-02-04
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    @Yudop Where do $f$ and $df/dx$ exist? Perhaps a better question is where are they not defined? Don't overthink it, the answer isn't complicated. – 2017-02-04
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    @Kaynex I read the Existence and uniqueness theorem, and since the functions are continous and contain the point $(to,xo)$ the interval where the solution exists is $-\infty < t < \infty$ right? – 2017-02-06
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    @Yudop Yep! And further more, the range of the function exists everywhere. – 2017-02-06

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