
Partial Diff erential Equations 8-43
Now, dz = p dx + q dy = p d(x + ay) (8.264)
2/3
2/3
2/3
5/3
()
1
3
5
1
dz a
d x ay z dz
p
a
a
xayc z
a
⎛⎞
+==
⎜⎟
⎝⎠
+
⎛⎞
++=
⎜⎟
⎝⎠
+
(8.265)
Standar
d Form III: Separable equation f (x, p) = g ( y, q).
We can take f (x, p) = a and g(y, q) = a (8.266)
Solving for p and q we get
p = f
1
(a, x), q = g
1
(a, y) (8.267), (8.268)
Substituting in dz = p dx + q dy = f
1
(a, x) dx + g
1
(a, y) dy (8.269)
Integrating we get the CI as
11
(,) (, ) zfaxdxgaydyc=++
∫∫
(8.270)
Example 8.11.7 Solve p + q + x + y.
Solution We can write the equation as
p − x = y − q = a (8.271)
so that
p = x + a, q = y − a (8.272)
Substituting in
dz = p dx + q dy (8.273) ...