Page 262 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 17: Lagrange’s Methods for Solving Partial Differential Equations

The general integral is obtained by eliminating between Notes
z = x + {exp. ( /a)} y + f ( )

y
and 0 = x {exp. ( /a)} + f (a)
a

Example 2: Find the complete integral of
2 2
2 2
x p + y q = z 2
Solution:
X
Z
Now put z = e , x = e , y = e Y
z z X z Y 1 z
p . . .
x X x Y x x X
z
xp = .
X
Z Z z 1 z
and now . .
X z X z X

Z
xp = z .
X
Similarly,

Z
yq = z
Y
The equation becomes

2 2
Z 2 Z 2
2
z z z
X Y
2 2
Z Z
or 1.
X Y
The complete integral is

Z = aX + bY + c
2
2
where a + b = 1
i.e., log z = a log x (1 a 2 ) log y c .

m
2m
n
2n
l
Example 3: p sec x + z q cosec y = z lm/(m n) .
Solution:
2
2
Put cos x dx = dX, sin y dy = dY and z 1/(m n) dz = dZ.
Write the given equation as
m n
z 1/(m n ) dz z 1/(m n ) z
. . 1
2
cos x dx sin x y
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