Page 251 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 17: Lagrange’s Methods for Solving Partial
Differential Equations

CONTENTS
Objectives
Introduction
17.1 Linear Partial Differential Equations of the First Order

17.2 Lagrange’s Method of Solutions
17.3 Illustrative Examples
17.4 Some Special Types of Equations
17.5 Summary
17.6 Keywords
17.7 Review Questions

17.8 Further Readings

Objectives

After studying this unit, you should be able to:
 Understand that Lagrange’s method involves one dependent variable and two or more
independent variables in the differential equation.
 See that in the method the technique involved is similar to that which occurs in total
differential equation.

 Know how to study some special methods of solving non-linear partial differential
equations.

Introduction

Lagrange’s method is quite suitable to linear differential equations involving more than two
independent variables.
Four different methods are also listed to deal with special types of differential equations.

17.1 Linear Partial Differential Equations of the First Order

z z
Let p and q .
x y
Then the linear partial differential equations involving z as dependent and x, y as independent
variables are of the form
Pp + Qq = R ... (1)
where P, Q and R are given functions of x, y and z and they do not involve p and q. The first
systematic theory of equations of this type was given by Lagrange. Equation (1) is frequently
referred to as Lagrange’s equation.

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