Show that the substitution φ − φ₀ = ψ e^{−βt/(cρA)} reduces the heat equation for a rod with lateral loss to ψ_t = a²ψ_xx

Exercise 1.2.3(b)

$Handwritten solution, page 1 of 1. Show that the substitution φ − φ₀ = ψ e^{−βt/(cρA)} reduces the heat equation for a rod with lateral loss to ψ_t = a²ψ_xx$