
128 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
10. a
∂w
∂x
+ b
∂w
∂y
= cw + cos
k
(λx) cos
n
(βy).
This is a special case of equation 1.4.7.13 with f (x) = cos
k
(λx) and g(y) = cos
n
(βy).
11. ax
∂w
∂x
+ by
∂w
∂y
= cw + k cos(λx + µy).
General solution:
w = x
c/a
k
a
Z
x
0
t
−(a+c)/a
cos
λt + µt
b/a
x
−b/a
y
dt + Φ
x
−b/a
y
.
12. x
∂w
∂x
+ y
∂w
∂y
= ax cos(λx + µy)w + b cos(νx).
General solution:
w = exp
ax
λx + µy
sin(λx + µy)
Φ
y
x
+ b
Z
cos(νx) exp
−
a
λ + µu
sin
(λ + µu)x
dx
,
where u = y/x. In the integration, u is considered a parameter.
13. a cos
n
(λx)
∂w
∂x
+ b cos
m
(µx)
∂w
∂y
= c cos
k
(νx)w + p cos
s
(βy).
This is a special case of equation 1.4.7.22 with f (x) = a cos
n
(λx), g
1
(x) ≡ 0,