Page 280 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 18: Charpit s Method for Solving Partial Differential Equations

dz = p dx + q dy Notes
c dx dy
=
[(1 c )(cx y )]

z (1 + c) = 2 (cx + y) 1/2 + b.

Example 5: Solve by Charpit s method:

pq = px + qy.
Solution:
The auxiliary equations are

dp dq dz dx dy
.
p q ( p x q ) q (y p ) (x q ) (y q )
From first two ratios,
p/q = a i.e., p = aq.

Putting the value of p in the given equation,
aq 2 = aqx + qy
or q = (y + ax)/a.

Therefore
p = (y + ax).
Now dz = p dx + q dy
y ax
= (y + ax) dx + dy.
a
adz = (y + ax) (dy + a dx).
2
az = (y + ax) /2 + c.
Writing c as f (a),
2
az = (y + ax) /2 + f (a). ... (1)
Differentiating with respect to a,
z = x (y + ax) + f (a). ... (2)
Eliminating a between (1) and (2) the general integral will be obtained.

Example 6: Solve by Charpit s method:
2
2zx px 2qxy + pq = 0.
Solution:

Applying Charpit s method,
dx dy dz dp dq df
.
x 2 q 2xy p px 2 2xyq 2z 2qy 0 0

q = a.

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