Page 323 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes x y 2
e x y
Now, P.I. = 2 2
(D D 2)(D D 1) D D 3D 2
2
e x y x y
=
(1 D 2)D D 3D D 2 D 2
2 1
2 2 2 2

2
e x y 1 D 3D D 2 D 2 D 3D D 2 D 2
= 1 0
4 2 2 2 2 2 2 2 2 2

3
D 3D D 2 D 2 2
... x
2 2 2 2

2
2
2
e x y 1 D 3D D 2 D 2 3DD 3D D 3D D 3D D 2
= 1 ... x .
4 2 2 2 2 4 2 2 4 8
x y 2
e 1 3x y 3 3
2
= x y xy y 3x 3
4 2 2 2 2 4
2
e x y x y 3x 2 3y xy 3x 21
=
4 2 4 4 2 2 8
The solution is
2
e x y x y 3x 2 3y xy 3x 21
z = e 2x (y ) x e x (y ) x
4 2 4 4 2 2 8

Example 4: Solve the equation:
2
x
(D 3 4D D 4DD 2 )u = cos(y 2 )
2
x
or D (D 2D ) u = cos ( 2 )
x
Solution: C.F. is ( ) (y 2 ) x (y 2 )
y
x
1 1 (y 2 )
x
x
P.I. = 2 cos (y 2 ) 2 sin ,
(D 2D ) D (D 2D ) 2
1 x
Now since ax by = (ax by ),
(bD aD ) b
x
x
1 1 sin(y 2 ) 1 x sin(y 2 )
P.I. =
(D 2D ) D 2D 2 (D 2D ) 2
x 2
x
= sin(y 2 )
4

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