
2.1. Equations of the Form f(x, y, z)
∂w
∂x
+ g(x, y, z)
∂w
∂y
+ h(x, y, z)
∂w
∂z
= 0 159
◮ Coefficients of equations contain logarithmic and power-law functions.
5.
∂w
∂x
+ ax
n
∂w
∂y
+ b ln
k
(λx)
∂w
∂z
= 0.
Integral basis: u
1
= y −
a
n + 1
x
n+1
, u
2
= z − b
Z
ln
k
(λx) dx.
6.
∂w
∂x
+
ay + c ln
k
(λx)
∂w
∂y
+
bz + s ln
n
(βx)
∂w
∂z
= 0.
Integral basis:
u
1
= ye
−ax
−c
Z
ln
k
(λx)e
−ax
dx, u
2
= ze
−bx
− s
Z
ln
n
(βx)e
−bx
dx.
7. ax
∂w
∂x
+ by
∂w
∂y
+
c ln
n
(λx) + s ln
k
(βy)
∂w
∂z
= 0.
This is a special case of equation 2.1.7.21 with f (x, y) = c ln
n
(λx) + s ln
k
(βy).
8. ax ln(λx)
∂w
∂x
+ by ln(βy)
∂w
∂y
+ cz ln(γz)
∂w
∂z
= 0.
Integral basis: u
1
= b ln
ln(λx)
− a ln
ln(βy)
, u
2
= c ln
ln(λx)
− a ln
ln(γz)
.
9. ax ln(λx