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Sachin Kaushal, Lovely Professional University Unit 8: Total Differential Equations, Simultaneous Equations

Unit 8: Total Differential Equations, Notes
Simultaneous Equations

CONTENTS
Objectives
Introduction

8.1 Total Differential Equation
8.2 Condition of Integrability of Total Differential Equation
8.3 Methods for Solving the Differential Equations
8.4 Simultaneous Differential Equations
8.5 Summary

8.6 Keywords
8.7 Review Questions
8.8 Further Readings

Objectives

After studying this unit, you should be able to:
 Deal with equations which are total differentials as well as simultaneous differential
equations involving more than one dependent variable and one independent variable.
 See whether total differential equations are integrable and study the condition of
integrability as well its uniqueness of the solution.

Introduction

The total differential equations are seen to be integrable with some illustrated examples. There
are four differential methods of obtaining the solution of total differential equations. The
conditions when the total differential is exact are obtained.

8.1 Total Differential Equation

An equation of the form
P dx Q dy R dz = 0 ...(i)
, ,
Where, P, Q, R are functions of x, y, z is known as total differential equation . The equation (i) is
said to be integrable if there exists a relation of the form
x
y
z
u ( , , ) = c, ...(ii)
which on differentiation gives the above differential equation (i). The relation (ii) is called the
complete integral or solution of the given differential equation.
Now consider equation (i). If (ii) is the integral of (i) and since

u u u
du = dx dy dz , ...(iii)
x y z

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