
98 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
4. a
∂w
∂x
+ b sinh
n
(λx)
∂w
∂y
=
c sinh
m
(µx) + s sinh
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 sinh
n
(λx),
and h(x, y) = c sinh
m
(µx) + s sinh
k
(βy).
5. a
∂w
∂x
+ b sinh
n
(λy)
∂w
∂y
=
c sinh
m
(µx) + s sinh
k
(βy)
w.
This is a special case of equation 1.3.7.19 with f (x) = a, g(y) = b sinh
n
(λy), h
1
(x) =
c sinh
m
(µx), and h
2
(y) = s sinh
k
(βy).
◮ Coefficients of equations contain hyperbolic cosine.
6. a
∂w
∂x
+ b
∂w
∂y
=
c cosh(λx) + k cosh(µy)
w.
General solution: w = exp
c
aλ
sinh(λx) +
k
bµ
sinh(µy)
Φ(bx − ay).
7. a
∂w
∂x
+ b
∂w
∂y
= c cosh(λx + µy)w.
General solution:
w =
exp