---
tag: 075VBY4K4X
kind: definition
name: Initial value problem
aliases: [cauchy-problem]
msc: [34A12]
defines: 0FXDW992DS
uses: [0P97V20RW7, 04SW7WECS8]
sources:
  - url: https://en.wikipedia.org/wiki/Initial_value_problem
    locator: Definition
---

## Definition

Let $\Omega$ be an open subset of $\mathbb{R} \times \mathbb{R}^m$, let $f : \Omega \to \mathbb{R}^m$ be a function, and let $(t_0, y_0) \in \Omega$.
The initial value problem for $f$ at $(t_0, y_0)$ is the system
$$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.$$
A solution of the problem is a [[solution|04SW7WECS8]] $y : I \to \mathbb{R}^m$ of the equation $y' = f(t, y)$ on an open interval $I$ with $t_0 \in I$, such that $y(t_0) = y_0$.
The condition $y(t_0) = y_0$ is the initial condition.

## Remarks

For an equation of order $n$ in explicit form, the initial conditions are the values $y(t_0), y'(t_0), \dots, y^{(n-1)}(t_0)$. The [[reduction to a first-order system|0TH9CPW3VS]] changes this problem into the problem above.
The problem $y' = y$, $y(0) = 1$ has the unique solution $y(t) = e^t$.
The [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique local solution when $f$ is continuous and Lipschitz in $y$. The [[Peano existence theorem|098YP831QY]] gives a solution when $f$ is only continuous.

## Explanations

### L1 human

A differential equation alone usually has many solutions.
The initial condition says where the solution starts, and this picks one solution.
Suppose that you know the speed of a car at each moment and its position at the start. Then you know its position at each moment.