Page 270 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 17: Lagrange’s Methods for Solving Partial Differential Equations

2
2
16. p q = pz. Notes
17. pz = 1 + q 2
18. p (1 + q) = qz.

Standard IV

If the equation is of the type
f (x, p) = f (y, q), ... (1)
1 2
write f (x, p) = f (y, q) = c ... (2)
1 2 1
Solving equations (2) for q and p, we have
z/ x = p = (x, c )
1 1
and z/ y = q = (y, c ).
2 1
Now dz = p dx + q dy
= (x, c ) dx + (y, c ) dy,
1 1 2 1
x
y
c
c
z = 1 ( , )dx ( , )dy b .
1
1
The general integral may be obtained from the above complete integral and as in Standard I,
there is no singular integral.
Illustrative Examples

Example 1: Find complete integral of:

x
p q 2 .
Solution:

p 2x q a (say),

2
2
p = (2x + a) and q = a ,
dz = p dx + q dy
2
2
= (2x + a) dx + a dy
(2x a ) 3
2
z = + a y + b
3.2
the complete integral is
2
3
6z 6b = (2x + a) + 6a y.
2
2
2
2
2
Example 2: Solve: z (p + q ) = x + y .
Solution:
2
Put z dz = dZ; i.e. Z = z /2.

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