
58 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
40.
∂w
∂x
+
f
′
x
g
y
2
−
g
′
x
f
∂w
∂y
= 0.
Principal integral:
Ξ =
1
f(fy + g)
+
Z
f
′
x
dx
f
2
g
.
41. f
2
∂w
∂x
+
f
′
x
y
2
− g(y − f)
∂w
∂y
= 0.
Principal integral:
Ξ =
f
2
E
y − f
+
Z
f
′
x
E dx, E = exp
−
Z
g dx
f
2
.
42.
∂w
∂x
+
y
2
−
f
′′
xx
f
∂w
∂y
= 0.
Principal integral: Ξ =
1
f(fy + f
′
x
)
+
Z
dx
f
2
.
43. g
∂w
∂x
+
afgy
3
+ (bf g
3
+ g
′
x
)y + cfg
4
∂w
∂y
= 0.
Principal integral:
Ξ =
Z
dv
av
3
+ bv + c
−
Z
fg
2
dx, v =
y
g
.
44.
∂w
∂x
+
fy
3
+ 3fhy
2
+ (g + 3f h
2
)y + f h
3
+ gh − h
′
x
∂w
∂y
= 0.
Principal integral:
Ξ =
E
(y + h)
2
+ 2
Z
fE dx, E = exp
2
Z
g dx
.
45.
∂w
∂x
+
hh
g
′
x
f
2
(ag + b)
3
y
3
+
f
′
x
f
y + f g
′
x
ii
∂w
∂y
= 0.
Principal integral:
Ξ =
Z
dv
v
3
−av + 1
−
1
a
ln |ag + b|, v =
y
f(ag + b)
.
46.
∂w
∂x
+
hh
(y − f)( ...