
1.1. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= 0 63
24.
hh
f
y
x
+ x
a
h
y
x
ii
∂w
∂x
+
hh
g
y
x
+ yx
a−1
h
y
x
ii
∂w
∂y
= 0.
Principal integral:
Ξ = x
−a
E + a
Z
h(v)E dv
g(v) − vf (v)
, v =
y
x
,
where E = exp
a
Z
f(v) dv
g(v) − vf (v)
.
25.
f(ax + by) + bxg(ax + by)
∂w
∂x
+
h(ax + by) −axg(ax + by)
∂w
∂y
= 0.
Principal integral:
Ξ = xE −
Z
f(v)E dv
af(v) + bh(v)
, v = ax + by,
where E = exp
−b
Z
g(v) dv
af(v) + bh(v)
.
⊙ Literature: V. F. Zaitsev and A. D. Polyanin (1996).
26.
f(ax + by) + byg(ax + by)
∂w
∂x
+
h(ax + by) −ayg(ax + by)
∂w
∂y
= 0.
Principal integral:
Ξ = yE −
Z
h(v)E dv
af(v) + bh(v)
, v = ax + by,
where E = exp
a
Z
g(v) dv
af(v) + bh(v)
.
27. x
f(x
n
y
m
) + mx
k
g(x
n
y