
1.1. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= 0 53
11.
∂w
∂x
+
fy
2
− ax
n
gy + anx
n−1
+ a
2
x
2n
(g − f)
∂w
∂y
= 0.
Principal integral:
Ξ =
E
y − ax
n
+
Z
fE dx, E = exp
a
Z
x
n
(2f − g) dx
.
12. x
∂w
∂x
+ (fy
2
+ ny + ax
2n
f)
∂w
∂y
= 0.
1
◦
. Principal integral for a > 0:
Ξ = arctan
y
√
ax
n
−
√
a
Z
x
n−1
f dx.
2
◦
. Principal integral for a < 0:
Ξ = arctanh
y
p
|a|x
n
+
p
|a|
Z
x
n−1
f dx.
⊙ Literature: V. F. Zaitsev and A. D. Polyanin (1996).
13. x
∂w
∂x
+
x
2n
fy
2
+ (ax
n
f − n)y + bf
∂w
∂y
= 0.
Principal integral:
Ξ =
Z
dv
v
2
+ av + b
−
Z
x
n−1
f dx, v = x
n
y.
◮ Equations contain arbitrary and exponential fu nctions.
14.
∂w
∂x
+ (ae
λx
y
2
+ ae
λx
fy + λf )
∂w
∂y
= 0.
Principal integral:
Ξ =
e
−2λx
E
ay + λe
−λx
+
Z
e
−λx
E dx, E = exp ...