Page 263 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes which on substitution becomes

Z m Z n
1.
X X
The complete integral is
Z = aX + bY + c

where
m
n
a + b = 1
m n (m n l )/(m n )
and Z .z
m n a
1 1
X (x sin2 ).
x
2 2
1 1
Y (y sin2 ).
y
2 2

2
Example 4: Solve: (y x) (qy px) = (p q) .
Solution:
Put x + y = X, xy = Y

z z X z Y
p . .
x X x Y x
z z
y
.1 . ;
X Y
z z X z Y
q . .
y X y Y y
z z
.1 . .
x
X Y
The given equation by this substitution becomes

z z z z
(y x ) x y y x
X Y X Y

2
z z z z
y x .
X Y X Y
2 2
z z
(y x ) 2 (y x ) 2
X Y
2
z z
or
X Y

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