Page 281 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Putting this value in the given equation,
2
2zx px 2axy + ap = 0.
2
p = 2x (z ay) / (x a).
Also dz = p dx + q dy

x
2 (z ay )
= dx + a dy
(x 2 ) a
dz a dy 2x
or = 2 dx
z ay x a
2
or log (z ay) = log c(x a).
2
(z ay) = c (x a).
2
z = ay + c(x a) is the general solution.

Example 7: Solve by Charpit s method:
2
2
p + q 2px 2qy + 1 = 0.
Solution:
Applying Charpit s method,

dp dq
F F F F
p q
x z y z
dp dq
i.e. i.e. p = qa.
2p 2q

Substituting in the given equation,
2
2
q (a + 1) 2q (ax + y) + 1 = 0.
2(ax y ) [4(ax y ) 2 4(a 2 1)]
q = 2 [taking +ve sign with the radical].
2(a 1)

(ax y ) (ax y ) 2 (a 2 1)]
q = 2
(a 1)

Now dz = p dx + q dy
1 1
2
2
= (ax + y) (a dx + dy) + [(ax + y) (a + 1)] (a dx + dy).
(a 1) (a 1)
Now putting ax + y = t
a dx + dy = dt
2
2
2
(a + 1) dz = dt + [t (a + 1)] dt.

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