2.2.26. Consider the transport equation + c(t, x) θα ди = Ot θα O with time-varying wave speed. dx = c(t, x), Define the corresponding characteristic ordinary differential equation to be dt the graphs of whose solutions x(t) are the characteristic curves. (a) Prove that any so- lution u(t, x) 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 k is an arbitrary constant. Prove that (t, x) 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 C¹.