Cauchy initial value problem

Let D be a subset of n×, (x0,t0) a point of D, and f:D be a function.

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

{x(t)=f(x(t),t)x(t0)=x0 (1)


  1. 1.

    x is a differentiable function x:In defined on a interval I;

  2. 2.

    one has (x(t),t)D for all tI and t0I;

  3. 3.

    one has x(t0)=x0 and x(t)=f(x(t),t) for all tI.

We say that a solution x:In is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution y:Jn defined on an interval JI and such that y(t)=x(t) for all tI, one has I=J (and hence x and y are the same function).

We say that a solution x:In is a global solution if D=n×I.

We say that a solution x:In is unique if given any other solution y:In one has x(t)=y(t) for all tI (i.e. x is the unique solution defined on the interval I).

0.1 Notation

Usually the differential equationMathworldPlanetmath in (1) is simply written as x=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=f(y,t) or y=f(y,x). It is also common to write x˙=f(x,t) when the independent variable represents a time value.

0.2 Examples

  1. 1.

    The function x(t)=logt defined on I=(0,+) is the unique maximal solution to the Cauchy problemMathworldPlanetmath:


    In this case f(x,t)=1/t, D={(x,t):t0}, t0=1, x0=0.

  2. 2.

    The function x(t)=et is a global (and hence maximal), unique solution to the Cauchy problem:

  3. 3.

    Consider the Cauchy problem


    The function x(t)=0 defined on I= is a global solution. However the function y(t)=t3 defined on I=[0,+) is also a solution and so are the functions

    z(t)={(t-c)3if tc0if t<c.

    for every c0. 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 problemMathworldPlanetmathPlanetmath
Related topic InitialValueProblem
Related topic DifferentialEquation
Related topic CauchyKowalewskiTheorem
Defines solution to the Cauchy problem
Defines solution to the initial value problem