Page 267 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes z dz X dz
q = . . a
y dX y dX
Therefore the equation becomes

2 2
dz 2 dz
9 z a 4
dX dX
2
dz 2
{9z 9 } 4
a
dX
dz 2
dX 3 (z a 2 )

2
2
or (z a )dz dY
3
)
(z a 2 3/2 2
or X C
(3/2) 3
2 3
or (z + a ) = (X + k) 2
2 3
2
or (z + a ) = (x + ay + k) .
3
3
Example 2: Find a complete integral of: p + q 3pqz = 0.
Solution:
Put z = f (x + ay) = f (X)
2 3
dz 3 dz dz dz
a 3a z 0
dX dX dX dX

dz 3
(1 a ) az
dX
dz dx
or 3
3az 1 a
1 X
log z 3 c
3a 1 a
3
or 3a (x + ay) + k = (1 + a ) log z.

2 2
Example 3: Find a complete integral of: q y = z(z px).
Solution:

dy
Put dY = , i.e. y = e Y
y

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