The partial differential equation
is known as the wave equation. It models the motion of a wave u(x, y, z, t) in R3 and was originally derived by Johann Bernoulli in 1727. In this equation, c is a positive constant, the variables x, y, and z represent spatial coordinates, and the variable t represents time.
(a) Let u = cos(x − t) + sin(x + t) − 2ez+t − (y − t)3. Show that u satisfies the wave equation with c = 1.
(b) More generally, show that if f1, f2, g1, g2, h1, and h2 are any twice differentiable functions of a single variable, then
satisfies the wave equation with c = 1.
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