Do not use Laplace Transforms for solution, although you may use them to check your work. (Ctrl) Problem #5 Demetrios opens an account with an initial investment Plo) of $5000. The annual interest rate, r, is 10%. (a) If the interest is compounded continuously and Demetrios makes an additional $2000 deposit, d, every year, what will be the balance at the end of 10 years? Show the differential equation and its implicit solution P(t) then solve this explicitly for P(10). (b) Let I represent the annual interest rate, m the number of compounding periods in a year, Po the initial investment, and d the fixed deposit at the end of each compounding period. Then the balance at the end of each compounding period is generated by the first order difference equation, 1 P(n) +d, P(0) = P. m Use the given hint below to show that the balance at the end of n compounding periods is given by, P(n+1) =( (1+) md) (1++" - md P(n) = Po + m Hint: Use the following relation to prove the above solution P(n) to the difference equation for P(n+1): If a(n + 1) = ra(n) +b, then the solution is: a(n) = (20 - ) pa + in (c) Given the solution P(n) to the difference equation for P(n+1) in part b, solve the following: If the interest is compounded monthly (12 times per year) and $250 is deposited at the end of each compounding period, what will be the balance after 10 years