
Chapter 2
First-Order Equations with Three
or More Independent Variables
2.1 Equations of the Form
f(x, y, z)
∂w
∂x
+ g(x, y, z)
∂w
∂y
+ h(x, y, z)
∂w
∂z
= 0
◆
For brevity, only an
integral basis
u
1
= u
1
(x, y), u
2
= u
2
(x, y)
of an equation will often be presented in Section 2.1. The general solution of the equation
is given by
w = Φ(u
1
, u
2
),
where
Φ = Φ(u
1
, u
2
)
is an arbitrary function of two variables.
2.1.1 Equations Containing Power-Law Functions
◮ Coefficients of equations are linear in x, y, and z.
1. a
∂w
∂x
+ b
∂w
∂y
+ c
∂w
∂z
= 0.
Integral basis: u
1
= bx −ay, u
2
= cx −az.
⊙ Literature: E. Kamke (1965).
2.
∂w
∂x
+ ax
∂w
∂y
+ by
∂w
∂z
= 0.
Integral basis: u
1
= ax
2
− 2y, u
2
= 3z + bx(ax
2
−