
Chapter 1
First-Order Equations
with Two Independent Variables
1.1 Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= 0
◆
For brevity, often only a
principal integral
Ξ = Ξ(x, y)
of an equation will be presented in Section 1.1. The general solution of the equation is
given by
w = Φ(Ξ),
where
Φ = Φ(Ξ)
is an arbitrary function.
1.1.1 Equations Containing Power-Law Functions
◮ Coefficients of equations are linear in x and y.
1. a
∂w
∂x
+ b
∂w
∂y
= 0.
General solution: w = Φ(bx − ay), where Φ is an arbitrary function.
⊙ Literature: E. Kamke (1965).
2. a
∂w
∂x
+ (bx + c)
∂w
∂y
= 0.
Principal integral: Ξ =
1
2
bx
2
+ cx −ay.
3.
∂w
∂x
+ (ax + by + c)
∂w
∂y
= 0.
Principal integral: Ξ = (abx + b
2