Answer:
(∂e(x, t) ∂t + ∂φ(x, t) ∂x - Q(x, t) ) dx = 0, so the integrand is zero, giving the same equation as before. ∂e(x, t) ∂t = - ∂φ(x, t) ∂x + Q(x, t), we obtain the heat equation c(x)ρ(x) ∂u(x, t) ∂t = ∂ ∂x ( K0(x) ∂u(x, t) ∂x ) + Q(x, t).
