Page 259 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 3 t
or log tan 2 logc z .
2
8 2
3 x y
z 2 tan . b
8 2
Hence the general solution is

3 x y
2
x y
[sin(x y ) cos(x y )]e z tan
8 2
Example 9: Solve:

t t t
(t y z ) (t z x ) (t x y ) x y . z
x y z
Solution:
The auxiliary equations are

dx dy dz dt
t y z t z x t x y x y z

dx dy dz dt dx dt (dt dt ) dz dt
or
3(x y z t ) (x t ) (y t ) (z t )
log (x + y + z + t) 1/3 = log c (x t)
1
log (x + y + z + t) 1/3 = log c (y t)
2
and log (x + y + z + t) 1/3 = log c (z t)
3
Hence the solution is
[x + y + z + t] 1/3 (x t), (x + y + z + t) 1/3 (y t), (x + y + z + t) 1/3 (z t)] = 0

Example 10: Solve:
z z z xy
x y t az .
x y t t

Solution:
The auxiliary equations are

dx dy dt dz
.
x y t xy
az
t
From (1) and (2),

log c x = log y, i.e., y = c x.
1 1
From (1) and (3), t = c x
2

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