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Example 2.6 To see another application of similarity variables, any partial differ-
ential equation of the form

u t u x
F tx, u, , = 0
x t
is equivalent to the ODE

du du
F ξ, u, , = 0
dξ dξ
where ξ = tx. Performing the change of variables,

1 ∂u 1 ∂ξ du 1 du du
= = x =
x ∂t x ∂t dξ x dξ dξ
1 ∂u 1 ∂ξ du 1 du du
= = t = .
t ∂x t ∂x dξ t dξ dξ

For example the partial differential equation

∂u x ∂u
2
u + + tx u = 0
∂t t ∂x
which can be rewritten
1 ∂u 1 ∂u
u + + txu = 0,
x ∂t t ∂x
is equivalent to
du du
u + + ξu = 0
dξ dξ

where ξ = tx.

Conclusion 2.1 The method of characteristics is a powerful method that allows one
to reduce any first-order linear PDE to an ODE, which can be subsequently solved
using ODE techniques. We will see in later lectures that a subclass of the second
order PDEs (second order hyperbolic equations) can be also treated with a similar
characteristic method.

Conclusion 2.2 Solutions of PDEs generally depend on arbitrary functions (instead
of arbitrary constants). You need an auxiliary condition if you want to determine a
unique solution. Such conditions are usually called initial or boundary conditions.
We shall encounter these conditions throughout the book.

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