Page 339 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2
or m bm (ac eh ) = 0 ...(3)
If m , m are the roots of (3), the first system of intermediate integrals is given by
1 2
U dy T dx U dp = 0,
1 1
U dx 2 R dy 2 U dq = 0,

e e
i.e., by edy c dx e dp = 0.
m 1 m 1

e e
edx ady e dq = 0.
m m
2 2
or by c dx e dp m dy = 0,
1
a dy e dq m dx = 0;
2
so one of the intermediate integrals is

cx ep m y = ( f ay eq m x ). ...(4)
2
1
Similarly the second intermediate integral is
(cx ep m y ) = F (ay ap m x ), ...(5)
1
1
It is not possible to get the values of p and q from (4), (5); so we combine (4) with cx ep m y , A
2
Thus we have
(m 2 m 1 )y A = ( f ay eq m x )
2

or ay eq = m x [(m 2 m 1 )y A ]
2
where is inverse function of f.

This gives q, and cx ep m y A gives p.
2
Substituting these values in dz p dx q dy ,
e dz = (A cx m y )dx [ ay m x {(m 2 m 1 )y A }]dy .
2
2
Integrating,

cx 2 ay 2
ez = m xy Ax { (m 2 m 1 )y A } B
2
2 2
t
f ( )dt
t
where ( ) = m m
2 3
Example 2: Solve:

2
z (1 q 2 )r 2pqzs z (1 p 2 )t z 2 (s 2 rt ) 1 p 2 q = 0.

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