
1.4. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h
1
(x, y)w + h
0
(x, y) 123
7. a
∂w
∂x
+ b
∂w
∂y
= c cosh
k
(λx)w + s cosh
n
(βx).
This is a special case of equation 1.4.7.1 with f (x) = cosh
k
(λx) and g(y) = cosh
n
(βx).
8. a
∂w
∂x
+ b
∂w
∂y
=
c
1
cosh
n
1
(λ
1
x) + c
2
cosh
n
2
(λ
2
y)
w
+ s
1
cosh
k
1
(β
1
x) + s
2
cosh
k
2
(β
2
y).
This is a special case of equation 1.4.7.16 in which f(x) = c
1
cosh
n
1
(λ
1
x), g(y) =
c
2
cosh
n
2
(λ
2
y), p(x) = s
1
cosh
k
1
(β
1
x), and q(y) = s
2
cosh
k
2
(β
2
y).
9. x
∂w
∂x
+ y
∂w
∂y
= ax cosh (λx + µy)w + b cosh(νx).
General solution:
w = exp
ax sinh(λx + µy)
λx + µy
Φ
y
x
+ b
Z
cosh(νx) exp
−
a sinh[(λ + µu)x]
λ + µu
dx
x
,
where u = y/x. In the integration, u is considered a parameter.