Problem 13.12 Water flows between the North American Great Lakes as depicted in Fig. P13.12. Based on mass balances, the following differential equations can be written for the concentrations in each of the lakes for a pollutant that decays with first-order kinetics: %\"l- - (0 0056+ kig s 001+ k) ?f't . K % %""tl- 0.01902 + 001387 - 0047+ kig, %- 033597 - 0376+ kg, %f_i = 011364, - 0133+ kig - FIGLRE P1312 The Marth American Great Lakes. The arrows Indicate how water flows between the lakes where k = the first-order decay rate (fyr), which is equal to 0.69315/(half-life). Note that the constants in each of the equations account for the flow between the lakes. Due to the testing of nuclear weapons in the atmosphere, the concentrations of strontium-90 (905r) in the five lakes in 1963 were approximately {e} = {17.7 30.5 43.9 136 3 30.1)7 in units of Bg/m?>. Assuming that no additional 905r entered the system thereafter, use MATLAB and the approach outlined in Prob. 13.11 to compute the concentrations in each of the lakes from 1963 through 2010. Note that 905r has a half-life of 28.8 years. Problem continued on next page. . Problem 13.11 A system of two homogeneous linear ordinary differential equations with constant coefficients can be written as - 5u+ 3yr (0) = 50 2ya = 100m- 3012, (0) = 100 at If you have taken a course in differential equations, you know that the solutions for such equations have the form yi = cient where c and A are constants to be determined. Substituting this solution and its derivative into the original equations converts the system into an eigenvalue problem. The resulting eigenvalues and eigenvectors can then be used to derive the general solution to the differential equations. For example, for the two-equation case, the general solution can be written in terms of vectors as (1 = qtyler+ qlule where full = the eigenvector corresponding to the ith eigenvalue ()) and the c's are unknown coefficients that can be determined with the initial conditions. (a) Convert the system into an eigenvalue problem. (b) Use MATLAB to solve for the eigenvalues and eigenvectors. (c) Employ the results of (b) and the initial conditions to determine the general solution. (d) Develop a MATLAB plot of the solution for t = 0 to 1