
12-18 Mathematical Methods
so that we have dz = a dx + f (a) dy.
Integrating, we get the complete integral as
() or ()zax fadyc zaxfayc=+ + =+ +
∫
(12.66)
where a and c are arbitrary constants.
Note 1 We can put q = a instead of p = a and proceed
to obtain the complete integral.
Example 12.41
Solve pq = k
Solution The complete integral is
k
zax yc
a
=+ +
(1)
where a and
c are arbitrary constants
The general integral is obtained by eliminating
a between the equations
1
()
k
zax y a
a
f=+ + (2)
where c has been replaced by f ( a) and
2
0()
k
xya
a
f=− +
′
(3)
obtained by differentiating with respect to a.
The singular integral, if it exists, is determined
from the ...