
1.3. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h(x, y)w 113
31. ax
∂w
∂x
+ by
∂w
∂y
= f (x, y)w.
General solution:
w = exp
1
a
Z
1
x
f
x, u
1/a
x
b/a
dx
Φ(u), where u = y
a
x
−b
.
In the integration, u is considered a parameter.
32. f (x)
∂w
∂x
+ g(x)y
∂w
∂y
= h(x, y)w.
General solution:
w = Φ(u) exp
Z
h(x, uG)
f(x)
dx
, where u =
y
G
, G = exp
Z
g
f
dx
.
In the integration, u is considered a parameter.
33. f (x)
∂w
∂x
+
g
1
(x)y + g
0
(x)
∂w
∂y
= h(x, y)w.
General solution:
w = Φ(u) exp
Z
h(x, uG + Q)
f(x)
dx
, u =
y − Q
G
,
where G = exp
Z
g
1
f
dx
and Q = G
Z
g
0
dx
fG
. In the integration, u is considered a
parameter.
⊙ Literature: A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux (2002).
34. f (x)
∂w
∂