Page 268 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

Notes
dx
X
and dX = , i.e. x = e ,
x
The equation becomes
2
z z
z z ,
Y X
z = f (X + aY) = f ( ).

2
2 dz dz
a z z
d d

2
2 dz dz 2
a z z 0.
d d
2 2
dz z (z 2 4a z )
d 2a 2

dz 1
or 2 2 d .
z [1 (1 4 )] 2a
a
2
[1 (1 4 ) 1]
a
log z c 1
2a 2
2
2
2a log z = [ (1 + 4a ) 1] [X + aY] + k
2
= [ (1 + 4a ) 1] (log x + a log y) + k.
m n l
Example 4: Find complete integral of: pq = x y z .
Solution:

x m 1 y n 1
Put X , Y ,
m 1 n 1

z z X z z Y
. , . ,
x X x y Y y
z m z z n
p x ,q y .
x X Y

z z l
The given equation becomes . z ,
X Y
which is of the form f (p, q, z) = 0.
z dz dz z
Putting , a ,
X d dy d

dz dz l
a ; z
d d

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