
48 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
24. x
∂w
∂x
+
λ(arccos x)
n
y
2
+ ky + λb
2
x
2k
(arccos x)
n
∂w
∂y
= 0.
Principal integral: Ξ = arctan
y
bx
k
− λb
Z
x
k−1
(arccos x)
n
dx.
◮ Coefficients of equations contain arctangent.
25.
∂w
∂x
+
a arctan
k
(λx) + b
∂w
∂y
= 0.
Principal integral: Ξ = y − bx − a
Z
arctan
k
(λx) dx.
26.
∂w
∂x
+
a arctan
k
(λy) + b
∂w
∂y
= 0.
Principal integral: Ξ = x −
Z
dy
a arctan
k
(λy) + b
.
27.
∂w
∂x
+ k arctan
n
(ax + by + c)
∂w
∂y
= 0.
Principal integral:
Ξ =
Z
dv
a + bk arctan
n
v
− x, v = ax + by + c.
28.
∂w
∂x
+ a arctan
k
(λx) arctan
n
(µy)
∂w
∂y
= 0.
Principal integral: Ξ = a
Z
arctan
k
(λx) dx −
Z
dy
arctan
n
(µy)
.
29.
∂w
∂x
+
y
2
+ λ(arctanx)
n
y − a
2
+ aλ(arctan x)
n
∂w
∂y
= 0.