Cauchy initial value problem
Let $D$ be a subset of ${\mathbb{R}}^{n}\times \mathbb{R}$, $({x}_{0},{t}_{0})$ a point of $D$, and $f:D\to \mathbb{R}$ be a function.
We say that a function $x(t)$ is a solution to the Cauchy (or initial value) problem
$$\{\begin{array}{cc}{x}^{\prime}(t)=f(x(t),t)\hfill & \\ x({t}_{0})={x}_{0}\hfill & \end{array}$$  (1) 
if

1.
$x$ is a differentiable function $x:I\to {\mathbb{R}}^{n}$ defined on a interval $I\subset \mathbb{R}$;

2.
one has $(x(t),t)\in D$ for all $t\in I$ and ${t}_{0}\in I$;

3.
one has $x({t}_{0})={x}_{0}$ and ${x}^{\prime}(t)=f(x(t),t)$ for all $t\in I$.
We say that a solution $x:I\to {\mathbb{R}}^{n}$ is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution $y:J\to {\mathbb{R}}^{n}$ defined on an interval $J\supset I$ and such that $y(t)=x(t)$ for all $t\in I$, one has $I=J$ (and hence $x$ and $y$ are the same function).
We say that a solution $x:I\to {\mathbb{R}}^{n}$ is a global solution if $D\subset ={\mathbb{R}}^{n}\times I$.
We say that a solution $x:I\to {\mathbb{R}}^{n}$ is unique if given any other solution $y:I\to {\mathbb{R}}^{n}$ one has $x(t)=y(t)$ for all $t\in I$ (i.e. $x$ is the unique solution defined on the interval $I$).
0.1 Notation
Usually the differential equation^{} in (1) is simply written as ${x}^{\prime}=f(x,t)$. Also, depending on the topics, the name chosen for the function and for the variable, can change. Other common choices are ${y}^{\prime}=f(y,t)$ or ${y}^{\prime}=f(y,x)$. It is also common to write $\dot{x}=f(x,t)$ when the independent variable represents a time value.
0.2 Examples

1.
The function $x(t)=\mathrm{log}t$ defined on $I=(0,+\mathrm{\infty})$ is the unique maximal solution to the Cauchy problem^{}:
$$\{\begin{array}{cc}{x}^{\prime}(t)=1/t\hfill & \\ x(1)=0.\hfill & \end{array}$$ In this case $f(x,t)=1/t$, $D=\{(x,t):t\ne 0\}$, ${t}_{0}=1$, ${x}_{0}=0$.

2.
The function $x(t)={e}^{t}$ is a global (and hence maximal), unique solution to the Cauchy problem:
$$\{\begin{array}{cc}{x}^{\prime}(t)=x(t)\hfill & \\ x(0)=1.\hfill & \end{array}$$ 
3.
Consider the Cauchy problem
$$\{\begin{array}{cc}{x}^{\prime}(t)=\frac{3}{2}\sqrt[3]{x}\hfill & \\ x(0)=0.\hfill & \end{array}$$ The function $x(t)=0$ defined on $I=\mathbb{R}$ is a global solution. However the function $y(t)=\sqrt{{t}^{3}}$ defined on $I=[0,+\mathrm{\infty})$ is also a solution and so are the functions
$$ for every $c\ge 0$. So there are no unique solutions. Moreover $y$ is not a maximal solution.
Title  Cauchy initial value problem 
Canonical name  CauchyInitialValueProblem 
Date of creation  20130322 14:57:18 
Last modified on  20130322 14:57:18 
Owner  paolini (1187) 
Last modified by  paolini (1187) 
Numerical id  14 
Author  paolini (1187) 
Entry type  Definition 
Classification  msc 34A12 
Synonym  Cauchy problem 
Synonym  initial value problem^{} 
Related topic  InitialValueProblem 
Related topic  DifferentialEquation 
Related topic  CauchyKowalewskiTheorem 
Defines  solution to the Cauchy problem 
Defines  solution to the initial value problem 