Existence-uniqueness

Consider an IVP for a general first order scalar ODE.

${\begin{cases}y'=f(t,y)\\y(t_{0})=y_{0}\end{cases}}$

The existence-uniqueness theorem theorem states that if $f(t,y)$ and its derivative w.r.t. y is continuous in some rectangle $a<t<b$, $c<y<d$ about $(t_{0},y_{0})$, then there exists a unique solution of the IVP defined for some time interval $a'<t<b'$ about $t_{0}$.

Notes

The time interval may be much shorter than $a<t<b$. The solution gives no information on the time interval. This means that the solution can go off to undefined values even when the inputs are defined.

Geometric intuition

Geometrically, $f(t,y)$ is the slope. Consider the vector $(1,f(t,y))$. This vector would be tangential to the movement of the curve.