
78 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
5. a
∂w
∂x
+ b ln
n
(λy)
∂w
∂y
= c ln
m
(µx) + s ln
k
(βy).
This is a special case of equation 1.2.7.20 with f (x) = a, g(y) = b ln
n
(λy), h
1
(x) =
c ln
m
(µx), and h
2
(y) = s ln
k
(βy).
6. a ln
n
(λx)
∂w
∂x
+ b ln
k
(βy)
∂w
∂y
= c ln
m
(γx).
General solution:
w =
c
a
Z
ln
m
(γx)
ln
n
(λx)
dx + Φ(u), where u = b
Z
dx
ln
n
(λx)
− a
Z
dy
ln
k
(βy)
.
◮ Coefficients of equations contain logarithmic and power-law functions.
7. a
∂w
∂x
+ b
∂w
∂y
= cx
n
+ s ln
k
(λy).
General solution: w =
c
a(n + 1)
x
n+1
+
s
b
Z
ln
k
(λy) dy + Φ(bx − ay).
8.
∂w
∂x
+ a
∂w
∂y
= by
2
+ cx
n
y + s ln
k
(λx).
This is a special case of equation 1.2.7.3 with f(x) = b, g(x) = cx
n
, and h(x) = s ln
k
(λx).
9.
∂w
∂x
+