Page 329 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Now it may be possible to get one or two relations between x, y, z, p, q called intermediate
integrals, and then to find the general solution of (1).
If (3) resolves into two linear equations in dx and dy such as

dy m dx = 0, and dy m dx 0, ...(5)
1 2
from one of the equations (5) combined with (4) and if necessary with dz pdx qdy , we may
obtain two integrals u 1 a and 1 ; b then u 1 f 1 ( 1 ),
where f is an arbitrary function, is an intermediate integral.
1
Proceeding similarly from the second equation, we may get another intermediate integral u =
2
f ( ).
2 2
From these two integrals we may find the values of p and q and putting these values in dz = p dx
+ q dy and integrating it we get the complete integral of the original equation.

Illustrative Examples

Example 1: Solve by Monge’s method r a 2 . t
Solution: (This can be easily solved by the method discussed in the last section. Here we solve it
by Monge’s Method).
dp s dy dq s dx
2
2
Putting r and t in the given equation, dpdy a dx dq ( s dy 2 a dx 2 ).
dx dy
So the subsidiary equations are
2
dy 2 a dx 2 = 0 ...(1)
2
and dp dy a dx dq = 0. ...(2)
From (1)

dy a dx = 0 ...(3)
dy a dx = 0. ...(4)

Taking (3) and combining with (2), we get

dp adq = 0.
p qa = A.

Also y + ax = B.
p aq (y ax is an intermediate integral.
)
)
Similarly p aq (y ax is the second intermediate integral.
From these,

1
p = [ (y ax ) (y ax )]
2

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