
244 FIRST-ORDER EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES
7.
∂w
∂x
+
y
2
− a
2
+ aλ sinh(λx) − a
2
sinh
2
(λx)
∂w
∂y
+ f (x) sinh(γz)
∂w
∂z
= g(x)w + h(x).
General solution: w = exp
Z
g(x) dx
Φ(u
1
, u
2
) +
Z
h(x) exp
−
Z
g(x) dx
dx
,
where
u
1
=
Z
f(x) dx −
1
γ
ln
tanh
γz
2
,
u
2
=
E
y − a cosh (λx)
+
Z
E dx, E = exp
2a
λ
sinh(λx)
.
◮ Coefficients of equations contain arbitrary functions of different variables.
8.
∂w
∂x
+ f (x)
∂w
∂y
+ g(y)
∂w
∂z
= aw + h(x).
General solution: w = e
ax
Φ(u, v) +
Z
e
−ax
h(x) dx
, where
u = y − F(x), v = z −
Z
x
x
0
g
u + F (t)
dt, F (x) =
Z
f(x) dx,
and x
0
may be taken as arbitrary.
⊙ Literature: A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux (2002).
9.
∂w
∂x
+ f (