V 1 V 2. r 3. r 6 r 4. Math 2250 Lab 12 Due Date : 4/25/2017 at 6:00pm

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1 Math 50 Lab 1 Name: Due Date : 4/5/017 at 6:00pm 1. In the previous lab you considered the input-output model below with pure water flowing into the system, C 1 = C 5 =0. r 1, C 1 r 5, C 5 r r V 1 V r 6 x 1 (t) r 4 x (t) The data from the previous lab was r 1 = r = r = r 4 =100gal/hrandr 5 = r 6 = 00gal/hr. As a result, V1(t) 0 =V(t) 0 =0,andwetookV 1 =50galandV =100gal. For this week s lab problem let C 1 = C =0.lb/gal, and keep the previous rate and volume data. apple x1 (t) (a) Use your modeling ability to verify that solves the non-homegeneous linear x (t) system of DE s apple x 0 1(t) x 0 (t) = apple 4 1 apple x1 (t) x (t) + apple (1)

2 (b) Find the general solution to (1). (Hint: x = x p + x h. For x p try a constant vector x p = d, with undetermined entries. You found x h last week.) (c) Solve the initial value problem for (1), assuming there are initially 9lbs of salt in tank 1 and 1lbs of salt in tank. Page

3 . [Network of Tanks] The following figure depicts a system of three connected tanks of varying volume. Salt water of concentration c 1,in =0(1+sin( t)) grams per liter flows into tank 1 from an external source at rate r 1,in =L/mandleavesthesystemfromtankatrate r,out = L/m. Additionally, salt water flows between the tanks at rate r i,j liters per minute, where i is the index of the tank the fluid flows into and j is the tank the fluid flows from. We assume that at time t =0thetanksarefull. Becausetheflowintoeachtankis exactly matched by the flow out of the tank, each tank remains full for all time. r 1,in =[L/m] c 1,in = 0(1 + sin( t)) x 1 (t) V 1 = 0 r,1 =1[L/m] r,1 =[L/m] x (t) V = 10 x (t) V = 10 r, =1[L/m] r,out =[L/m] (a) Let x(t) = 4 x 1 x x 5, wherex(t) denotesthevectorofsaltquantitiesineachtank. Write the system of di erential equations described by this system in the form x 0 (t) =Ax(t)+c(t) where A is a by matrix. Page

4 (b) Find the eigenvalues and eigenvectors of A. (c) Find the homogeneous solution x H (t) (calledthecomplementarysolutionx c (t) in the text), i.e. the solution to x 0 (t) =Ax(t) Hint: It will be a linear combination of functions of the form e t v. Page 4

5 (d) The vector c(t) inthenon-homogeneoussystemtellsusthatthereisaparticular solution (x p )ofthissystemwhichhastheform: x p (t) =w 1 + w cos ( t)+w sin ( t). Plug this formula into the di erential equation and solve for the (undetermined coe cient) vectors using matrix algebra. w 1, w,andw. Hint: Use technology to find the decimal entries for w, w. (e) Write the general solution to the nonhomogeneous system of di erential equations. Page 5

6 . [Carbon Dioxide vibrations] In this problem, we want to investigate the vibrational properties of carbon dioxide, CO, where we model the interaction between molecules as Hookean springs. (This is the linearized model, valid for small oscillations.) The model looks as follows: where k is the spring constant, and x 1,x,x are the displacements of the atoms from their equilibrium locations, as indicated. (a) Denote the mass of the carbon atom and oxygen atoms by m C =1andm O =16 respectively. (These are their atomic weights.) Suppose the two springs that connect the atoms have the same spring constant k = 96, in appropriately chosen units. Use Newton s Second Law and the example above to derive a system of three second order di erential equations for x 1 (t),x (t),x (t). Page 6

7 (b) Rewrite the system in part (a) in the matrix form where x 00 = Ax, x 1 x = 4x 5 x is the vector of displacements. Hint: Mathematically this is similar to the -car train system in Example of the text section 7.4. (c) Find the eigenvalues and eigenvectors of the acceleration matrix A. Page 7

8 (d) Write down the general solution of the system in part (b). You might want to refer to Theorem 1 in the textbook, page 470. (e) Describe the two fundamental modes of vibration, and the one fundamental mode of translation, associated to the solution in part (c). You can see these two longitudinal vibration fundamental modes illustrated, along with two transverse oscillation modes, at the website Page 8

9 4. [Optional matrix exponential extra credit problem] Matrix exponentials are useful in finding solutions to homogeneous linear systems of first order di erential equations (when the matrix (A) isconstant). Moreprecisely,ifA is an n n matrix, then the (unique) solution of the initial value problem x 0 = Ax, x(0) = x 0 is given by x(t) =e At x 0. In this question, we will learn how to compute e At for the case when A is a diagonalizable matrix. (The text covers matrix exponentials in detail, in Chapter 8.) Recall that an n n matrix A is said to be diagonalizable if there exists an invertible matrix P whose columns are linearly independent eigenvectors of A such that AP = P, where is the diagonal matrix which has the corresponding eigenvalues in the diagonal entries. We can also write this identity as A = P P 1. (a) Show that for any positive integer k, A k = P k P 1. (b) Show that if = n 7 5 then k = 6 4 k 1 k k n Page 9

10 (c) The exponential of a square matrix is defined using the infinite sum e A = 1X k=0 1 k! Ak Thus, e At = 1X k=0 1 k! (At)k = 1X k=0 t k k! Ak Use this together with results from part (a) and (b) to show that e At = P 6 4 e 1 t e t P 1 e nt Page 10

11 (d) As shown above, we can compute the matrix exponential if we know the matrix P whose columns are linearly independent eigenvectors. More precisely, if j is an eigenvalue of A with its corresponding eigenvector v j,then P =[v 1 v... v n ], = n Note that the eigenvectors in P are in the same order as the eigenvalues in. To illustrate all of your work, consider the initial value problem (IVP) dx dt = x +8y dy dt =x + y with initial condition x(0) = 1, y(0) = 1. Rewrite the problem in the matrix form apple x x 0 = Ax, where x = y. Compute the matrix exponential e At.Thenverifythatx(t) =e At x(0) is the solution to the IVP. Page 11