l. Consider the transport equation �� +c(t, x) �� = O with time-varying wave speed c(t, x).
Define the corresponding characteristic ordinary differential equation to be �; = c(t, x), the
graphs of whose solutions x(t) are the characteristic curves.
(a) lf u(t, x) satisfies to the partial differential equation �� + c(t, x) �� = O and x(t) is the
solution of t = c(t, x); and let F(t) = u(t, x(t)). Prove that a¡; = O.
(b) Suppose that the general solution to the characteristic equation t = e( t, x) is written
in the form �(t, x) = k, where k is an arbitrary constant. Prove that u(t, x) =
F(�(t, x)) is a solution to the time-varying transport equation �� + c(t, x)�� = O for
any continuously differentiable scalar function F E C1
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