# The Cauchy Problem and Wave Equations

Chapter

## Abstract

In the theory of ordinary differential equations, by the initial-value problem we mean the problem of finding the solutions of a given differential equation with the appropriate number of initial conditions prescribed at an initial point. For example, the second-order ordinary differential equation
and the initial conditions
constitute an initial-value problem.

$$
\frac{{d^2 u}}
{{dt^2 }} = f\left( {t,u\frac{{du}}
{{dt}}} \right)
$$

$$
u\left( {t_0 } \right) = \alpha , \left( {\frac{{du}}
{{dt}}} \right)\left( {t_0 } \right) = \beta ,
$$

## Keywords

Wave Equation Cauchy Problem Cauchy Data Cylindrical Wave Progressive Wave
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Birkhäuser Boston 2007