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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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