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Differential and Integral Equation
Notes
2 2
x
= x (3y 5 )
(D 3D )
= 2 x 2 {3(k 3 ) 5 }dx , where k + 3x = y
x
x
= 2 x 2 (3k 14 )dx
x
= 2kx 3 7x 4 2x 3 (y 3 ) 7x 4
x
= x 3 (2y x ).
Self Assessment
11. Solve
2
(D D ) z 2cosy x sin y
12. Solve
(D 2 DD 2D 2 )z (y 1)e x
20.6 The Non-homogeneous Equation with Constant Coefficients
The simplest case is
(D mD )z = 0
y
or z = e (mD ) a x ( )
where D has been considered algebraic and is arbitrary.
= e ax (y mx ).
Note. Also
(D mD )z = 0.
or p mq = z.
The subsidiary equations are
dx dy dz
= .
1 m z
z = e x (y mx ).
Similarly the integral of
(D m D 1 )(D m D 2 )(D m D 3 )... = 0
3
1
2
is z e 1 n 1 (y m x ) e 2 n 2 (y m x ) e 3 n 3 (y m x ) ...
1
2
2
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