Page 423 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Again Integrating
x t
y(x) = dt G ( )dt c c
t
1 2
0 0
x
t x
t
t
= t G ( )dt t G ( )dt c x c 2
1
0 0
0
x x
= x G ( )dt tG ( )dt c x c 2
t
t
1
0 0
x
t
G
or y(x) = (x t ) ( )dt c x c 2 ...(iv)
1
0
For x = 0, equation (iv) gives
0 = y(0) = 0 + c 0 + c or c = 0
1 2 2
Now (iv) becomes
x
y(x) = (x t ) ( )dt c x ...(v)
G
t
1
0
For t = 1
1
G
y(1) = 0 (1 t ) ( )dt c 1 .1
t
0
1
or c = (1 t ) ( )dt
G
t
1
0
Now equation (v) becomes
x 1
G
t
y(x) = (x t ) ( )dt x (1 t ) ( )dt
G
t
0 0
x 1
G
G
t
= (x t ) ( )dt (xt x ) ( )dt
t
0 0
x x 1
G
t
t
G
= (x t ) ( )dt (xt x ) ( )dt (xt x ) ( )dt
G
t
0 0 x
x 1
G
t
t
= (x 1) G ( )dt ( x t 1) ( )dt
0 x
1
x
t
t
or y(x) = K ( , ) ( )dt ...(vi)
G
0
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