
18 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
86. y
∂w
∂x
+
x
n−1
[(1 + 2n)x + an]y − nx
2n
(x + a)
∂w
∂y
= 0.
Principal integral:
Ξ = (x
n+1
+ ax
n
− y)
−1/n
+
Z
dv
a − v
−n
, v = x(x
n+1
+ ax
n
− y)
−1/n
.
87. y
∂w
∂x
+
[a(2n + k)x
k
+ b]x
n−1
y −(a
2
nx
2k
+ abx
k
−c)x
2n−1
∂w
∂y
= 0.
Principal integral:
Ξ = x
−k
E − ak
Z
E dv
nv
2
− bv −c
, v = x
−n
y − ax
k
,
where E = exp
−k
Z
v dv
nv
2
− bv − c
.
88. x(2axy + b)
∂w
∂x
−
a(m + 3)xy
2
+ b(m + 2)y − cx
m
∂w
∂y
= 0.
Principal integral: Ξ = cx
m+2
cx
m
− 2(m + 1)y(axy + b)
.
89. x
2
(2axy + b)
∂w
∂x
− (4ax
2
y
2
+ 3bxy − cx
2
− k)
∂w
∂y
= 0.
Principal integral: Ξ = (cx
2
+ k)
2
− 4cx
3
y(axy + b).
90. ax
m
∂w
∂x
+ by
n
∂w
∂y
= 0.
1
◦
. Principal integral for m 6= 1 and n 6= 1:
Ξ = b(