Page 321 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Illustrative Examples

2 z 2 z z z
Example 1: Solve: 2 2 3 3 xy e x 2y .
x y x y

)
Solution: Here, (D 2 D 2 3D 3D z xy e x 2y
or, (D D )(D D 3)z xy e x 2y

The complementary function is
(x ) y e 3x (y x ).

xy e x 2y
Now P.I. =
(D D )(D D 3) (D D )(D D 3)

1 e x 2y
= xy
D D (1 D )(1 D 3)
3(D D ) 1
3
1 (D D ) (D D ) 2 e x .e 2y
= 1 ... xy
3(D D ) 3 9 ( 1)(D 2)

1 x y 2 x 2y 1
= xy e .e .1
3(D D ) 3 3 9 D

1 x y 2 x 2y
= xy ye
D 3 3 9
3D 1
D
1 D D 2 x y 2 x 2y
= 3D 1 D D 2 ... xy 3 3 9 ye

1 x y 2 x 2 x x 2y
= xy ye
3D 3 3 9 2 3

2
x y 1 x 2 x 2x x 3 x 2
= ye x 2y
3.2 9 2 9 27 18 18
The solution is
2
x y x 2 xy 2x x 2 x 2y
3x
z = (x y ) e (y ) x ye
6 9 9 27 18
Example 2: Solve: (D D 1)(D D 2)z e 2x y . x
Solution: The complementary function is

e x (y ) x e 2x (y ) x

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