# 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 $f\colon D\to\mathbb{R}$ be a function.

We say that a function $x(t)$ is a solution to the Cauchy (or initial value) problem

 $\begin{cases}x^{\prime}(t)=f(x(t),t)\\ x(t_{0})=x_{0}\end{cases}$ (1)

if

1. 1.

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

2. 2.

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

3. 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 $x\colon I\to\mathbb{R}^{n}$ is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution $y\colon J\to\mathbb{R}^{n}$ 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 $x\colon I\to\mathbb{R}^{n}$ is a global solution if $D\subset=\mathbb{R}^{n}\times I$.

We say that a solution $x\colon I\to\mathbb{R}^{n}$ is unique if given any other solution $y\colon I\to\mathbb{R}^{n}$ 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. 1.

The function $x(t)=\log t$ defined on $I=(0,+\infty)$ is the unique maximal solution to the Cauchy problem:

 $\begin{cases}x^{\prime}(t)=1/t\\ x(1)=0.\end{cases}$

In this case $f(x,t)=1/t$, $D=\{(x,t)\colon t\neq 0\}$, $t_{0}=1$, $x_{0}=0$.

2. 2.

The function $x(t)=e^{t}$ is a global (and hence maximal), unique solution to the Cauchy problem:

 $\begin{cases}x^{\prime}(t)=x(t)\\ x(0)=1.\end{cases}$
3. 3.

Consider the Cauchy problem

 $\begin{cases}x^{\prime}(t)=\frac{3}{2}\sqrt[3]{x}\\ x(0)=0.\end{cases}$

The function $x(t)=0$ defined on $I=\mathbb{R}$ is a global solution. However the function $y(t)=\sqrt{t^{3}}$ defined on $I=[0,+\infty)$ is also a solution and so are the functions

 $z(t)=\begin{cases}\sqrt{(t-c)^{3}}&\text{if t\geq c}\\ 0&\text{if t

for every $c\geq 0$. So there are no unique solutions. Moreover $y$ is not a maximal solution.

 Title Cauchy initial value problem Canonical name CauchyInitialValueProblem Date of creation 2013-03-22 14:57:18 Last modified on 2013-03-22 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