
28 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
41. x
∂w
∂x
+
ax
2n
e
λx
y
2
+ (bx
n
e
λx
− n)y + ce
λx
∂w
∂y
= 0.
Principal integral:
Ξ =
Z
dv
av
2
+ bv + c
−
Z
x
n−1
e
λx
dx, v = x
n
y.
42. y
∂w
∂x
+ e
λx
(2aλx + a + b)y − e
λx
(a
2
λx
2
+ abx − c)
∂w
∂y
= 0.
Principal integral:
Ξ = xE +
Z
vE dv
λv
2
− bv −c
, v = e
−λx
y − ax,
where E = exp
a
Z
dv
λv
2
− bv −c
.
43. ae
λx
∂w
∂x
+ by
m
∂w
∂y
= 0.
1
◦
. Principal integral for m 6= 1: Ξ =
1
b(1 − m)
y
1−m
+
1
λa
e
−λx
.
2
◦
. Principal integral for m = 1: Ξ =
1
b
ln y +
1
λa
e
−λx
.
44. (ae
y
+ bx)
∂w
∂x
+
∂w
∂y
= 0.
1
◦
. Principal integral for b 6= 1: Ξ = xe
−by
−
a
1 −b
e
(1−b)y
.
2
◦
. Principal integral for b = 1: Ξ = xe
−y
− ay.
45. (ax
n
e
λy
+ bxy
m
)
∂w
∂x
+ e
µy
∂w
∂y
= 0.
Principal integral:
Ξ =
1
1 −