Consider a one-dimensional bar stretching from z = 0 to z = L and let u(x, t) be the temperature of the bar at position z at time t. The temperature obeys the heat equation, t z where a is some diffusion constant. (a) Perform a separation of variables u(x, t) = X(x)T(t) on the heat equation to get two ordinary differential equations - one only involving time and one only involving space.