Chapter 6 Partial Differential Equations

44 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS
or
We take
Then
− ln(x) =i ln(y)+K, K a constant.
α = ϕ 1 (x, y) = ln(x), β = ϕ 2 (x, y) = ln(y).
ã = x 2 ( 1
x 2 )
+2b 1 x (0)+0=1=˜c,
˜b =0,
(
˜d = x 2 − 1 )
+0+y 2 (0) = −1,
x 2 ( ) −1
ẽ = x 2 (0)+0+y 2 = −1.
y 2
Thus we have
u αα + u ββ − u α − u β =0.
We conclude this discussion by considering another example. The main point of this
example is to illustrate that the classification of a differential equation is a local result.
Example 6.3.9. The Tricomi Equation.
u yy − yu xx =0.
For this equation, D = b 2 − ac = y. So when y0
the equation is hyperbolic and when y = 0 the equation is parabolic.
For this example a = −y, b = 0 and c = 1 so the characteristic equation
dy
dx = b ± √ b 2 − ac
a
reduces to
√
dy −ac
dx = ± a
= ± 1 √ y
.
Since
For y>0 the equation is hyperbolic and the characteristic curves are are given by
3x ± 2y 3/2 = constant