20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS
1
1
2
y
x
− 1
x=1
c=1
y=c/x^2
Figure 20.2 The characteristics of equation (20.42). The shaded region shows
where the solution to the equation is defined, given the imposed boundary
condition atx= 1 betweeny=0andy= 1, shown as a bold vertical line.
P
y
Q
C
R
x
Figure 20.3 A boundary curveCthat crosses characteristics more than once.
20.6.2 Second-order equations
The concept of characteristics can be extended naturally to second- (and higher-)
order equations. In this case let us write the general second-order linear PDE
(20.19) as
A(x, y)
∂^2 u
∂x^2
+B(x, y)
∂^2 u
∂x∂y
+C(x, y)
∂^2 u
∂y^2
=F
(
x, y, u,
∂u
∂x
,
∂u
∂y
)
. (20.43)