Page 304 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 20: Higher Order Equations with Constant Coefficients and Monge’s Method

20.1 Linear Partial Differential Equations of Order n with Constant Notes
Coefficients; Complementary Functions

So far we have been dealing with partial differential equations of first order with first degree as
well as with any degree. In this unit we shall introduce higher derivatives than the usual first
z y 2 z 2 z 2 z
order derivatives , . So we may have 2 , , 2 and so on and so forth. If we are
x z x x y y
2 z 2 z 2 z
dealing with only second order equations we denote r 2 ,s and t 2 . In dealing
x x y y
with higher derivatives let us denote by D and by D , then
x y

2 2
D 2 , DD D D , D 2 ,...
x 2 x y y 2

n n 1
... D n , D n 1 D and so on. So we have to deal with a general equation of the form
x n x n 1 y

z z 2 z 2 z 2 z n z
x
y
y
F x , , , , , , , ,... ,... = f ( , ) ...(1)
z
x y x 2 x y y 2 x n
n
n
or A D z A D n 1 D z A D n 2 D 2 ... A D z
0
n
1
2
2
B D n 1 z B D n 2 D Z B D n 3 D z ... B n 1 D n 1 z
1
2
0
... M Dz M D z N z = f ( , ) ...(2)
y
x
1
0
0
Thus equation (1) may be written as
y
x
D
F ( ,D )z = f ( , ) ...(3)
Just as in the case of ordinary differential equations it can be shown that the complete solution
of linear partial differential equation will consist of two parts, namely:
(i) The complementary function (C.F.), and
(ii) The particular integral (P.I.)
The complementary function is the general solution of the equation
D
F ( ,D )z = 0 ...(4)
The particular integral is that value of z in terms of x, y which satisfies the equation (3) that
contains no arbitrary constants.
A Linear Homogeneous partial differential equation of order n with constant coefficients is that
)
)
in which ( ,F D D is a homogeneous function i.e. ( ,f D D and is of the form
f ( ,D )z = (A D n A D n 1 D ... A D n )z f ( , ) ...(5)
D
y
x
n
0
1
Non-homogeneous differential equation is not homogeneous i.e. if all terms of D, D in the
function F(D, D ) are not of the same degree.
LOVELY PROFESSIONAL UNIVERSITY 297