In a large slab of thickness l the temperature p(x,t) at a distance x from one face satisfies the partial differential equation \frac{\partial \varphi}{\partial t}=\kappa \frac{\partial^{2} \varphi}{\partial x^{2}} where k

is a positive constant. The faces x = 0 and x =l are both insulated (so that dp/dx = 0 on the faces) and at t = 0 the temperature for 0 < x < l is pox/l,where po is a constant. Use the method of separation of variables to show that y takes the form \varphi(x, t)=\frac{A_{0}}{2}+\sum_{n=1}^{\infty} A_{n} \cos \left(\frac{n \pi x}{l}\right) e^{-m^{2} \pi^{2} t / t^{2}} Determine the coefficients Ao and An (n= 1, 2,.). Write down the value of lim p(x, t) and give a brief physical interpretation.