Page 343 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes From (a),
p + ay = F ( ) ...(c)

q ax =
and from (b),

p ay = F ( ) ...(d)
q ax =
i.e., the two intermediate integrals are

p + ay = ( f q ax ) ...(1)

and p ay = F (q ax ) ...(2)

Now since it is not possible to find the values of p and q from (1) and (2), we proceed as follows.
Suppose , are not constants, but parameters.
Solving (c) and (d),

b
x = ,q . ...(3)
2a 2
1
F
p = [ ( ) f ( )], ...(4)
2
1
F
y = [ ( ) f ( )]. ...(5)
2a
Substituting these values in dz p dx q dy ,

1
F
F
dz = [ ( ) f ( )] (d dx ) [ ( )d f ( )d ]
4a 4a
1
f
F
= [{ ( )d F ( )d } { ( )d F ( )d }]
4a
1 1
F
f
F
[{ ( )d F ( )d } { ( )d f ( )dB }] [2 ( ) d 2 ( )d ].
f
4a 4a
1 2 2
z = [ F ( ) f ( ) f ( ) F ( )] f ( ) d F ( )d
4a 4a 4a
1 2 2
F
= [ ( ))( ) f ( )( )] G ( ) ( )
4a 4a 4a
F ( ) f ( ) 1 1
= G ( ) ( )
2a 2a 2a
or z qy = 1 (q ax ) 2 (q ax ) [from (3) and (5)]

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