Page 305 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Just as we deal with ordinary differential equation
x
(D n a D n 1 a D n 2 ... a n )y = f ( )
1
2
d
Where D , we shall deal briefly with the corresponding equation in two independent
dx
variables,
y
x
(D n a D n 1 D a D n 2 D 2 ... a D n ) z = f ( , ) ...(6)
n
1
2
where D and D .
x y
The simplest case is
(D mD )z = 0

i.e m z = 0
x y

or (p mq ) = 0

z z
where p = and q
x x
or z = (y mx )

This suggests what is easily verified, that the solution of (6) if ( , ) 0f x y is

Z = 1 (y m x ) 2 (y m x ) ... (y m x ) ...(7)
n
1
2
n
where the constants m m m 3 ,...,m are the roots (supposed all different)
,
,
1
n
2
m n a m n 1 a m n 2 .... a n = 0 ...(8)
2
1
Example: Solve
3 3 3
z 3 z 2 z = 0
2
x 3 x y x y 2
2
or (D 3 3D D 2DD 2 )z = 0
Now the roots of
m 3 3m 2 2m = 0

or 0, 1 and 2. So the solution is
x
y
z = F 1 ( ) F 2 (y x ) F 3 (y 2 )

298 LOVELY PROFESSIONAL UNIVERSITY