Show that if u is a solution of the heat equation ut = a²urr + F(x,t), then so is u+c1x + c2 for any choice of the constants c₁ and c2. Electric current in a long, insulated cable: Suppose we have an insulated wire with current i = i(x, t) and voltage E = E(x, t). Let R be the resistance, L be the inductance, C be the capacitance and G be the conductance (or leakage), all per unit length and all constant, of the wire. Then, if we look at a differential element of the wire from x to x + Ax, the potential drop along this element gives us ді Ət -AE= i(x, t)RAx + L(x, t)Ax. Also, the capacitance and inductance lead to ДЕ -Ai = GEAx + C(x, t)Ax. Ət a) Show that letting Ax → 0 gives us the two first-order PDEs: Ex + Ri + Li₁ = 0, ix + GECE₁ = 0. b) Differentiate the first equation by x and the second equation by t, then, along with the second equation above, eliminate i, and ixt to arrive at ExxLCEtt (RC + LG)Et + RGE. c) Instead, differentiate the first equation in part (a) by t and the second equation by x, then, along with the first equation in (a), eliminate Et and Ext to arrive at ixxLCitt (RC + LG) it + RGi. = Thus, E and i both satisfy the telegraph equation. d) If the inductance and leakage are very small and can be neglected, show that E and i both satisfy the heat/diffusion equation.