
2.3. Equations of the Form f
1
∂w
∂x
+ f
2
∂w
∂y
+ f
3
∂w
∂z
= f
4
w, f
n
= f
n
(x, y, z) 219
9.
∂w
∂x
+
f
1
(x) + f
2
(x)e
λy
∂w
∂y
+
g
1
(x) + g
2
(x)e
βz
∂w
∂z
= h(x)w.
The transformation
ξ = e
−λy
, η = e
−βz
, W = w exp
−
Z
h(x) dx
leads to an equation of the form 2.1.7.5:
∂W
∂x
− λ
f
1
(x)ξ + f
2
(x)
∂W
∂ξ
− β
g
1
(x)η + g
2
(x)
∂W
∂η
= 0.
◮ Coefficients of equations contain arbitrary functions of different variables.
10. f (x)
∂w
∂x
+ g(y)
∂w
∂y
+ h(z)
∂w
∂z
=
ϕ(x) + ψ(y) + χ(z)
w.
General solution:
w = exp
Z
ϕ(x)
f(x)
dx +
Z
ψ(y)
g(y)
dy +
Z
χ(z)
h(z)
dz
Φ(u
1
, u
2
),
where
u
1
=
Z
dx
f(x)
−
Z
dy
g(y)
, u
2
=
Z
dx
f(x)
−
Z
dz
h(z)
.
11. f (x)
∂w
∂x
+ z
∂w
∂y
+ g(y)
∂w
∂z
=
h
2
(x) + h
1
(y)
w.
General solution:
w = exp
Z
h
2
(x)
f(x)
dx +
Z
y
y
0
h
1
(t) dt
p