Math 2280-2, Practice Final ExamApril 22, 2008Total: 165/150 pointsProblem 1 (30 pts) Consider the three tank system depicted below.The tanks have capacities V1= 20 gal, V2= 50 gal and V3= 20 gal and they are assumedto be full at t = 0 min. The solution flow rate in the pipes between the tanks is r = 10gal/min. Let xi(t) be the quantity (pounds) of salt dissolved in the i−th tank at time t.Initially (t = 0 min) tank 1 contains 18 pounds of salt while the two other tanks containfreshwater.(a) Derive the first order system satisfied by the system of tanks. Put the system in theform x0= Ax, where x(t) = (x1(t), x2(t), x3(t))T.(b) Find the eigenvalues and eigenvectors of A.(hint: the eigenvalues are 0 and −3/5 ± i3/10)(c) Find the quantity of salt in each tank as a function of time, with the given initialcondition (hint: it is quicker not to compute the matrix exponential in this case)(d) What is the limiting quantity of salt in the tanks?Problem 2 (25 pts) Solve the following initial value problem using the Laplace transform(note: in the real final you will be provided an extract of the Table of Laplace Transformsin the front cover of your book)(hint: x(t) = (6e2t− 5 − e−3t)/15)(x(3)+ x00− 6x0= 0,x(0) = 0, x0(0) = x00(0) = 1.1Problem 3 (25 pts) Consider the logistic population modeled by the differential equationx0= 3x − x2(a) What are the critical points of the solution?(b) Find the population x(t) knowing that x(0) = x0.(c) Sketch typical solution curves for this population.(d) What is the limiting population if x0> 0?(e) Sketch the phase diagram for the population.Clearly specify the stability of each critical point.Problem 4 (30 pts) Consider the two population systemx0= 3x − x2−xy2y0= 4y − 2xy(a) How does each population behave in the absence of the other one (logistic, exponentialgrowth or decrease)?(b) Describe the nature of the interaction between the x−population and the y−population(competition, cooperation, predation)(c) Find the three critical points of the population (hint: one of them is (2, 2)T, and youessentially found the other two in the previous problem).(d) Classify the type and stability of each critical point.(e) Sketch the phase portrait of the population system. (note: if the point is a saddlepoint, carefully plot the axis of the hyperbolic trajectories, if the point is a spiral, findthe orientation by e.g. computing the tangent field at a couple of points)Problem 5 (30 pts) Consider a spring mass system governed by the DEx00(t) + 25x(t) = F (t),where F (t) is a 2π-periodic even external force with F (t) = t for 0 < t < π.(a) Sketch the function F (t) for t ∈ [−π, 3π].(b) Show that the Fourier series of F (t) isF (t) =π2−4πXnoddcos ntn2.(c) Use the Laplace transform method to find the steady periodic (particular) solution tox00+ 25x = cos 5t.What can you say about the magnitude of the solution? How is this phenomenoncalled?2(d) Use Fourier series and your previous answer to find the steady periodic solution tox00(t) + 25x(t) = F (t).Problem 6 (25 pts) Solve the boundary value problem (1D heat equation):∂u∂t= 2∂2u∂x2, for 0 < x < 1 and t > 0,u(0, t) = u(1, t) = 0, for t > 0,u(x, 0) = 5 sin πx −15sin 3πx, for 0 < x <