4: Consider the following One-Dimensional Heat Equation for heat flow in a rod of length L: Ut = Kuxx ux (t, 0) = 0 u(t, L) = To u(0, x) = To ( —— (L)2 where To is a fixed temperature. Find a series representation for u (in- cluding explicit formulas for all coefficients in terms of K, To and L).

The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is given by A(t)=a(e)rt, where a is the amount ofprincipal initially deposited into an account thatcompounds continuously. Prove that the percentageof interest earned to principal at any time t can becalculated with the formula I(t)=ert1.