The partial differential equation is known as the wave equation. It models the...

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) − 2e^z+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.