Page 265 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Differentiating with respect to a and b,
1
0 = x . b ,
2 a

a
0 = y ,
2 b

b a
x and y
a b
Eliminating a and b, the singular integral is
xy = 1.

2
2
Example 2: Solve z px qy = c (1 + p + q ).
Solution:
The complete integral is
2
2
z = ax + by + c (1 + a + b ) ... (1)
Differentiating with respect to a and b,
ca
0 = x , ... (2)
(1 a 2 b 2 )

bc
0 = y . ... (3)
(1 a 2 b 2 )

c 2 (a 2 b 2 )
2
x + y 2 = 2 2 .
1 a b
c 2 (a 2 b 2 )
2
2
2
c x y 2 = c 2 2
1 a b
c 2
= 2 2 .
1 a b
c 2
2
1 + a + b 2 = 2 2 2 .
c x y
Putting in (2), (3),

x (1 a 2 b 2 ) x
a = 2 2 2
c (c x y )

y
and b = 2 2 2 .
(c x y )

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