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6 6

c
∗
c = y ∗ b = y ∗
∗
y 2 3 2
b
y 3
a = y ∗
∗
y 1 1
d a
- -

d ∗

e
e ∗ f ∗
f

Figure 1 Figure 2

Example 2.1 Find the solution of the equation

4u x − 3u y = 0

satisfying the condition
u(0, y) = cos y.
From the above discussion we know that u will depend only on η = −5x − 4y, so
u(x, y) = f(−3x − 4y). The solution also has to satisfy the additional condition (called
initial condition), which we verify by plugging in x = 0 into the general solution:
cos y = u(0, y) = f(−4y).
z

So f(z) = cos − , and hence
4

3x + 4y
u(x, y) = cos ,
4
which one can verify by substituting into the equation and the initial condition.

2.2 Transport Equation

A particular example of a first order constant coefficient linear equation is the transport
equation
u t + cu x = 0, (2.5)

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