4. (DuZ, pp.161-162) Variable Changes. The Telegraph Equation (see Bell Chapter 7), given by
$$u_{tt} - c^2 u_{xx} + \delta u_t + m^2 u = 0,$$
is an example of the wave equation with all possible lower order terms. Consider
$$u_{tt} = a_1 u_{xx} + a_2 u_x + a_3 u_t + a_4 u.$$
(1)
(a) Introduce the variable change u(x,t) = exp(ax + ẞt)v(x, t) and plug this form into (1). Simplify
and present the resulting PDE.
(b) Since the coefficients a, ẞ are arbitrary, show that you can choose them (based on ai, i = 1, 2, 3, 4)
to eliminate the lower order terms involving ut and ux.
(There is nothing special about which two lower order terms we eliminate. However, whatever
lower order term is left, one then has to solve that equation....)
