Page 287 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
z z
(a) The equations involving only p and q . In this case the equation to be solved
x y
will be of the type
f (p, q) = 0 ... (1)
From the subsidiary equations

dp dq dz dx dy df
f f f f = f f f f 0 ... (2)
p q p p
x z z z p q p q

dp dq dz dx dy
or = ... (3)
0 0 f f f f
p q
p q q q
Now from first equation
dp = 0
or p = a = constant ... (4)
Substituting this value of p in (1) we have
f (a, q) = 0 ... (5)

Solving for q from (5) we have
q = (a) ... (6)
So from the equation

dz = p dx + q dy = a dx + (a) dy ... (7)
We have on integration
z = ax + (a) y + b
which is the general solution.

Example 1: Solve:
pq = 1
Solution:

1
Here again p = a so q =
a
Thus on integrating

dz = pdx + q dy
1
= a dx + dy
a

1
z = ax + y + b where a, b are constants
a

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