Existence-uniqueness

The existence-uniqueness theorem generally describes that a unique solution to IVP exists. It has varying (but very similar) criterias for different ODEs.

Linear second order

Suppose that p, q, and g are continuous functions over an interval $a<t<b$, and $y_{0},y_{0}'$ are real numbers. There exists a unique twice continuously differentiable function such that

${\begin{cases}y''+p(t)y'+q(t)y=g(t)\\y'(t_{0})=y_{0}'\\y(t_{0})=y_{0}\end{cases}}$

Non-linear first order

Consider an IVP for a general first order scalar ODE.

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

The theorem states that if $f(t,y)$ and ${\frac {\partial f}{\partial y}}$ 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.

Differentiability is important to ensure that the solution is unique.

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.