Page 422 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 24: Integral Equations and Algebraic System of Linear Equations

Notes
x 3
5 4 x
y
x
t
or y ( ) y (0) ( 8x 8t 2) ( )dt = x x
12 2
0
x 3
5 4 x
or y ( ) ( 8x 8t 2) ( )dt = x x 2 ...(8)
y
t
x
12 2
0
2
d y
Note: In this problem we have two answers, i.e. one for 2 another for y for the same
dx
problem, but they lead to the same conclusion.
Self Assessment
1. Express the differential equation

2
d y x dy x y ( ) 1 x with the condition at x = 0, y(0) = 4, dy 2, into integral
2
x
dx 2 dx dx x 0
equation.
24.3 Fredholm Integral Equations and Boundary Value Problem

Let us consider the following example of a second order differential equation with the given
boundary conditions and establish the integral equation

Example 1: Express the differential equation
2
d y ay ( ) = 0,
x
dx 2
with the boundary conditions
x = 0, y(0) = 0, x = 1, y(1) = 0,
as an integral equation
Solution: We have

2
d y
ay ( ) = 0 ...(i)
x
dx 2
with y(0) = 0 = y(1) ...(ii)
Method 1: Let
2
d y
dx 2 = G(x)

Integrating, we get

x
dy
t
= G ( )dt c ...(iii)
dx 1
0

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