Page 293 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 19: Jacobi’s Method for Solving
Partial Differential Equations

CONTENTS
Objectives
Introduction

19.1 Jacobi’s Method of Solution of Partial Differential Equations
19.2 Simultaneous Partial Differential Equations
19.3 Summary
19.4 Keywords
19.5 Review Questions

19.6 Further Readings

Objectives

After studying this unit, you should be able to:

 Know that Jacobi’s method for solving partial differential equation is similar to that of
Charpit’s method.

 See that two additional equations are to be found through which the first order derivatives
z z z
, , can be found that help in finding the solution of the first order partial
x x x
1 2 3
differential equations.

Introduction

Jacobi’s method consists of setting up the subsidiary equations.
Through the solution of subsidiary equations two independent integrals will be found and the
method uses techniques to solve the first order partial differential equation.

19.1 Jacobi’s Method of Solution of Partial Differential Equations

In Jacobi’s method we have to deal with three or more independent variables and one dependent
variable. Consider the equation

,
,
,
F x 1 , x x p p p 3 = 0 (1)
,
2
3
1
2
Where the dependent variable z does not occur except by its partial differential coefficients p , p ,
1 2
p with respect to the three independent variables x , x , x . The basic idea of Jacobi’s method is
3 1 2 3
very similar to that of Charpit’s.
So we try to find two additional equations
F x ,x ,x p p p =  ...(2)
,
,
,
1 1 2 3 1 2 3 1
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