Let f(x, y) = u(x, y, 0) be the initial temperature in a thin square plate of side π with edges kept at 0°C and faces perfectly insulated. Separating variables, obtain from u_t = c²∇²u the solution

Find the temperature in Prob. 31 if

f(x, y) = x (π - x)y(π - y).

Data from Prob. 31

Let f(x, y) = u(x, y, 0) be the initial temperature in a thin square plate of side π with edges kept at 0°C and faces perfectly insulated. Separating variables, obtain from u_t = c²∇²u the solution

where