
1.3. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h(x, y)w 103
7. a
∂w
∂x
+ b
∂w
∂y
=
c cos(λx) + k cos(µy)
w.
General solution: w = exp
c
aλ
sin(λx) +
k
bµ
sin(µy)
Φ(bx − ay).
8. x
∂w
∂x
+ y
∂w
∂y
= ax cos(λx + µy)w.
General solution: w = exp
ax
λx + µy
sin(λx + µy)
Φ
y
x
.
9. a
∂w
∂x
+ b cos
n
(λx)
∂w
∂y
=
c cos
m
(µx) + s cos
k
(βy)
w.
This is a special case of equation 1.3.7.33 with f (x) = a, g
1
(x) ≡ 0, g
0
(x) = b cos
n
(λx),
and h(x, y) = c cos
m
(µx) + s cos
k
(βy).
10. a
∂w
∂x
+ b cos
n
(λy)
∂w
∂y
=
c cos
m
(µx) + s cos
k
(βy)
w.
This is a special case of equation 1.3.7.19 with f (x) = a, g(y) = b cos
n
(λy), h
1
(x) =
c cos
m
(µx), and h
2
(y) = s cos
k
(βy).
◮ Coefficients of equations contain ...