Page 240 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Sachin Kaushal, LPU Unit 16: Classifications of Integrals of the First Order Partial Differential Equations

Unit 16: Classifications of Integrals of the First Order Notes
Partial Differential Equations

CONTENTS
Objectives

Introduction
16.1 Geometrical Theorems
16.2 Classes of Integrals of a Partial Differential Equation
16.3 General Integrals
16.4 Singular Integrals

16.5 Summary
16.6 Keyword
16.7 Review Questions
16.8 Further Readings

Objectives

After studying this unit, you should be able to:

 Know various methods of finding the solution of the first order partial differential equation.
 See that the solution may consists of two arbitrary constants and this type of solution is
called complete integral of the solution.

 Come to know that there are solutions which can be written in terms of an arbitrary
function. Such a solution is called a general integral. There is a typical solution also that is
called a singular solution.

Introduction

The types of integrals can be complete integrals that depend upon two arbitrary constants.
There is a general integral of the solution of partial differential equation that is expressed in
terms of one arbitrary constant or function.
Then there is a singular integral which is an other solution of the partial differential equation.

16.1 Geometrical Theorems

In this unit we shall be concerned mainly with equations of geometrical interest and seek the
solutions of various partial differential equations as integrals of various forms, general integrals,
complete integrals, particular integrals and singular integrals and their geometrical
interpretation.
For this purpose it is advisable to revise the following two geometrical theorems.
Theorem 1: The direction-cosines of the normal to the surface f(x, y, z) = 0 at the point (x, y, z) are
in the ratio

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