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Du 2.2.26. Consider the transport equation at + c(t. I) = 0 with time-varying wave speed. Define the corresponding characteristic ordinary differential equation to be ~ = c(t, x). dt the graphs of whose solutions r(t) are the characteristic curves. (a) Prove that any so- lution u(t, r) to the partial differential equation is constant on each characteristic curve. (b) Suppose that the general solution to the characteristic equation is written in the form {(t, x) = k, where & is an arbitrary constant. Prove that (t, ) defines a characteristic vari- able, meaning that u(t, x) = f(((t, x)) is a solution to the time-varying transport equation for any continuously differentiable scalar function fe Cl