Question

(7 10 Two connected tanks, each with a capacity of 60 liters, contain brine (salt water) Initially the firs tank contains 40 liters and the second tank contains 30 liers. Brine with a salt conccntration of 1 grams per litcr flows into the first tank at S litcrs per hour. Well-stirred brine flows from the first tank into the second at 7 liters per hour from the second nto the fist at3 lters per honr, from the first into a drain at 2 liter pcr hour, and from the sccond into a drain at 5 liters per hour. Initially thcrc arc 27 grams of salt in the first tank and 19 grams in the second tank (a) [8 Give an initial-valne problem that governs the grams of salt in each tank (b) 2 Give the interval of detinition for the solution of this initial-value problem. (S) 6What answer be produced by the following MATLAB command? >> B-12 3: 5 이·vect, val-eigisym(B)) You do not have to give the answer in MAILAB format (9) 22 A real 2x 2 matrix C has the eigenpairs 0)na(o()) and 3 (a) [4 Give a general solution to the linear planar system x Cx (b) 4] Give an invertible matrix V and a diagonal matrix D that diagonalizc C. (c) S Compute e'C. (d) Sketch a phase-plane portrait for this system and identify its type. Classify the stability of the origin. Carefully mark all sketched orbits with arrows! (10) 22 Solve the initial-value problem x - Ax, x(0) -x for the following A and x' A- [11) 22 Consider the system (a) [11] Find a nonconstant function H(2.3) 811ch that every orbit of the system satisfies , yc for s constantc (b) [11] The stationary points are (0, 2), (0, 2), and 1,0) In the phasc-planc sketch these stationary points plus the level set H(r,cfor each value of c that corresponds to astationary point that s a saddle. (No arrows are required!) (12) 22 Consider the system (a) 4] This system has two stationary points. Find then (b 6Find the Jacobian matrix at each stationary point (c) [6] Classify the type nd sしょbility ol each słalionary point (d) [6] Sketch a phase-plane portrait of the system that shows its behavior near each stationary point. (Carcfully mark all sketched orbits with arrows!) Final Exam, Math 246/246H Wednesday, 12 December 2018 Closed book. No electronics. Answer only one cquestion on each answer sheet Write your name and which question is being answered on each answer sheet Sign the Honor Pledge on the first answer sheet only. Indicate your answer to each part of each question lealy. Tndicate on the front of an answer sheet if your work continues on the back. Cross out work that you do not want considered Give rcasoning that justifics your answers (1) 22 Find an cxplicit solution to cach of the following initial-valuc problems. Idcntify their intervals of demiしion 0)-2 t2 - 4 (2) 10 Consider the following Matlab commands l'he following questions need not be answered in Matlab format! (a) [2] What dilTerenlial equation is being solved numerically? (b) 3 Give the intial condition for each solution being approximated. (c) l Over what time interval are the solutions being approximated? (d) [4] Sketch each ol these solutions over this ime interval on a single graph. Label the initial value of cach solution clearly (3) 22 Give an explicit real-valued general solution to cach of the following equations (1) 10 Given that t and solve the associated homogeneous differential equation, solve the iniiul-value proble Évaluate any definite integrals that arise (5) 22 Let y(1) be the solutio11 of the iniblal-value problem where fit)-1 + U(t-1)(el-z-1). Here u is the unit step function (a) [11] Find the Laplace transform Fs) of the forcing f () (b) 1 Find the Laplace transform Y(s) of the solution y(t) (DO NOT take the inverse Laplace transform to find yt) just solve for Yis)!) You may refer to the table e 4 s 23) (s3(s 17) (f6) 10 Find the function rt) whose Laplace transform is X(s) You шау reler to the table More Problems on the Other Side!

Answer 1

s+23 -3-구 はナ3 5-1チ ほt at、 himem,

Auui 66t 32-S lt

CA - 2D Cos 32 32 The 2 7.6G t lo 9.91ノー0-11

m3 m2 e. The US)

St1 e (s+ .465+3ヶ