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Sachin Kaushal, Lovely Professional University Unit 15: Cauchy’s Problem and Characteristics for First Order Equations

Unit 15: Cauchy’s Problem and Characteristics for Notes
First Order Equations

CONTENTS

Objectives
Introduction
15.1 Cauchy’s Problem for First Order Equations

15.2 Cauchy’s Method of Characteristics
15.3 Summary

15.4 Keywords
15.5 Review Questions
15.6 Further Readings

Objectives

After studying this unit, you should be able to:

 See that in the differential equation p and q may be of any degree also.

 Understand whether the solution exists for certain types of conditions or not.
 Understand that the partial differential equations can be solved by introducing certain
characteristic curves.

Introduction

The method of solution involves the ideas of integral surfaces or curves through which the
solution passes.

Thus one can introduce certain parameters and set up the characteristic equations for x, y, z, p and
q in terms of these parameters. After solving these equations and eliminating the parameters we
can get the solutions.

15.1 Cauchy’s Problem for First Order Equations

We know that z is a dependent variable and x, y being independent variables. So the first order
partial differential equation can be put into the form
(x, y, z, p, q) = 0 ...(1)

z z
Here p = and q = are partial derivatives. We are interested in seeking the solution of the
x y
partial differential equation (1). Before we attempt to find a solution we want to understand
whether the solution exists or not. What is meant by the existence theorem which establishes
conditions under which we can assert whether or not a given partial differential equation has
a solution at all. Also further whether the solution if it exists is unique or not. The conditions
to be satisfied in the case of first order partial differential equation are boiled down to the

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