Page 256 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

2
2
2
log (x + y + z ) = log z + log c Notes
2
2
2
2
(x + y + z ) = c z.
2
The solution is
y
x 2 y 2 z 2 z .
z
Example 5: Solve: (y + z) p + (z + x) q = (x + y).
Solution:
The auxiliary equations are

dx dy dz
.
y z z x x y

dx dy dz dx dy dy dz
2(x y z ) (x y ) (y z )

1
or log(x + y + z) = log c (x y)
2 1
and log (x y) = log c (y z)
2
Hence the solution is
x y
(x y) (x + y + z) = f .
y z

3
3
3
3
4
4
Example 6: Solve: (y x 2x ) p + (2y x y)q = 9z(x y ).
Solution:
The auxiliary equations are
dx dy dz .
3
3
y x 2x 4 2y 4 x y 9z x 3 y 3
3
dy 2y 4 x y .
3
dx y x 2x 4
dy dv dv 2v 4 v
Put y , x x , v x 3 .
dx dx dx v 2

dv 2v 4 v v 4 2v
x 3
dx v 2

v 2 2 dx
or 4
v v x

LOVELY PROFESSIONAL UNIVERSITY 249