
1.2. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h(x, y) 83
7. a
∂w
∂x
+ b
∂w
∂y
= c arccos(λx + βy).
1
◦
. General solution for aλ + bβ 6= 0:
w =
c
aλ + bβ
h
(λx + βy) arccos(λx + βy) −
p
1 −(λx + βy)
2
i
+ Φ(bx − ay).
2
◦
. General solution for aλ + bβ = 0:
w =
c
a
x arccos(λx + βy) + Φ(bx − ay).
8. x
∂w
∂x
+ y
∂w
∂y
= ax arccos(λx + βy).
General solution: w = ax
arccos(λx + βy) −
p
1 −(λx + βy)
2
λx + βy
+ Φ
y
x
.
9. a
∂w
∂x
+ b arccos
n
(λx)
∂w
∂y
= c arccos
m
(µx) + s arccos
k
(βy).
This is a special case of equation 1.2.7.35 with f(x) = a, g
1
(x) ≡0, g
0
(x) = b arccos
n
(λx),
and h(x, y) = c arccos
m
(µx) + s arccos
k
(βy).
10. a
∂w
∂x
+ b arccos
n
(λy)
∂w
∂y
= c arccos
m
(µx) + s arccos
k
(βy).
This is a