Existence of ODE solutions

Description

When do solutions to Ordinary differential equations exist? It turns out, almost always

Existence of solutions (Peano)

Let $D\subseteq {\mathbb{R}}^{n+1}$ be an open set and $f:D\to \mathbb{R}$ a Continuous function. Then every Initial value problem $x^{\prime }=f(t,x),x(t_0)=x_0$ has a local solution, i.e. there is an $ϵ>0$ s.t. the initial value problem has at least on solution on the intervall $[t_0-ϵ,t_0+ϵ]$.

Note: Higher-order equations are equivalent to higher-dimension equations. Look at the example below. What the theorem is saying is that, assuming one can bring the ODE into the form $x^{\prime }=f(t,x)$ and continuity is preserved, then the solution exists. This doesn’t need to be the case!

Global uniqueness and existence theorem

States there is a maximal interval on which the solution exists and is unique. See Uniqueness of ODE solutions for details

Example

Existence failing

Examint the following differential equation $x{y}^{\prime }=1,y(0)=0$ This looks like it would have a solution, but notice that the very similar equation $y^{\prime }=\frac{1}{x}$ is not so well defined at the starting point anymore.