Page 424 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 24: Integral Equations and Algebraic System of Linear Equations

Notes
(x 1)t if t x
with K(x, t) = ...(vii)
( x t 1) if t x
Using this in (i),

1
x
x
t
G
t
G ( ) a K ( , ) ( )dt = 0
0
2
t
d y (x 1) , t x
x
t
where G(x) = 2 , K ( , )
x
dx (t 1) , t x
Method 2:
Integrating (i) from 0 to x
x x
t
t
y ( )dt a y ( )dt = 0
0 0
x
x
y ( ) a y ( )dt = 0
t
t
0
0
t
t
x
or y ( ) y (0) a y ( )dt = 0
0
Again integrating,
x
x x
t
t
t
y
y ( ) y (0)[ ] a (x t ) ( )dt = 0
0 0
0
x
y
or y ( ) y (0) y (0)x a (x t ) ( )dt = 0
x
t
0
x
or y ( ) y (0)x a (x t ) ( )dt = 0 ...(viii)
x
t
y
0
Putting x = 1, this gives
1
y (1) y (0) a (1 t ) ( )dt = 0
y
t
0
or as y(1) = 0, we have
1
y (0) = a (1 t ) ( )dt
y
t
0

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