Page 255 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
2
2
Example 3: Solve: (z 2yz y ) p + (xy + zx) q = xy zx.
Solution:
The auxiliary equations are
dx dy dz
z 2 2yz y 2 xy zx xy zx

xdx ydy zdz
or 2 2 2 2
xz 2xyz xy xy xyz xyz z x

x dx + y dy + z dz = 0.
2
2
2
x + y + z = c .
1
Also from second and third terms,
dy dz
y z y z
or y dy z dy y dz z dz = 0
or y dy z dz (z dy + y dz) = 0

2
2
or y /2 z /2 yz = c .
2
The general solution is
2
2
2
2
2
(x + y + z , y z 2yz) = 0.
2
2
2
Example 4: Solve: (y z x ) p 2xyq + 2zx = 0.
Solution:
The auxiliary equations are
dx dy dz .
y 2 z 2 x 2 2xy 2zx
From second and third terms,
dy dz y
e
i
, . ., c 1 .
y z z
2x dx 2y dy 2z dz
Also 2 2 3 2 2 .
2xy 2xz 2x 4xy 4xz

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

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