Page 320 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

In case of repeated factors Notes
2
(D mD ) z = 0 ...(1)
or (D mD )(D mD )z = 0

let (D mD )z ,

Then, (D mD ) [from (1)]

or = e ax (y mx )

or (D mD )z = e ax (y mx );
1

y
)}
z = e (mD )x {e (mD )x e ax 1 (y mx dx ( )
ax
y
y
= e ( mD ) x ( )dx e e (mxD ) 2 ( )
= e x .x 1 (y mx ) e x 2 (y mx )
r
Similarly proceeding in the case of (D mD ) z 0, we have
x
x r
x
z = e x (y mx ) e x (y mx ) e x 2 (y mx ) ... e x 1 (y mx )
1 3 r
The Particular Integral
The methods for obtaining particular integrals of non-homogeneous partial differential
equations are very similar to those used in solving linear equation with constant coefficients.
Note: It can be easily shown that

1 ax by e ax by
I. e
F ( ,D ) F ( , )
D
b
a
provided ( , ) 0.F a b
1
II. sin(ax by ) or cos(ax by )
F ( ,D )
D
is obtained by putting D 2 a 2 , DD ab and D 2 b 2 , provided the denominator is
not zero.

1 m n 1 m n
F
D
III. x y [ ( ,D )] x y
F ( ,D )
D
which can be evaluated after expanding [ ( ,F D D )] 1 in ascending powers of D or D .
1 ax by
V
IV. (e . )
D
F ( ,D )
1
ax by
e .V
F {(D a ).(D b )}

LOVELY PROFESSIONAL UNIVERSITY 313