Page 1 of 12Student Name:Student Net ID:UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDEPARTMENT OF MATHEMATICSMATH 286 SECTION G1 – Introduction to Differential Equations PlusFINAL EXAMINATIONDECEMBER 20, 2012INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam is three (3) hours long. No personal aids or calculators are permitted.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.• EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.• GOOD LUCK!PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Question: 1 2 3 4 5 6 7 8 TotalPoints: 8 8 10 12 12 10 18 22 100Score:SOME USEFUL FORMULAS:eA=∞Xk=01k!Ak= I + A +12!A2+13!A3+ . . .x(t) = Φ(t)Φ(a)−1x(a) + Φ(t)ZtaΦ(s)−1f (s)dscos(a) cos(b) =12(cos(a + b) + cos(a − b))sin(a) sin(b) =12(cos(a − b) − cos(a + b))sin(a) cos(b) =12(sin(a + b) + sin(a − b)).Student Net ID: MATH 286 G1 Page 2 of 121. (8 points) Find an explicit solution to the initial value problemdydx= xexy2+ xex; y(0) = 0.Student Net ID: MATH 286 G1 Page 3 of 122. (a) (3 points) Prove that the differential equation1 + y3+ y cos(xy) +3y2x + x cos(xy)dydx= 0,is exact.(b) (5 points) Find an implicit solution to the ODE in part (a).Student Net ID: MATH 286 G1 Page 4 of 123. The town of Abnormal, Illinois is infected with a population of zombies. Suppose thatthe zombie population P (t) grows by infecting healthy humans, while at the same timeis partially “harvested” (i.e., destroyed) by the local health authority. Assume that P (t)is modeled by the “harvested logistic equation”dPdt= (15 − P )(P − 5) (thousands of zombies per day).(a) (6 points) Sketch the slope field for this differential equation.(b) (2 points) If at day 0, the zombie population is 10 thousand, what will the long-term population P∞= limt→∞P (t) of zombies be?(c) (2 points) For what range of initial populations P (0) will the zombie populationP (t) be eventually reduced to zero by the health authority?Student Net ID: MATH 286 G1 Page 5 of 124. (a) (6 points) Find the general solution to the differential equationy000+ y00+ 3y0− 5y = 0.(Hint: r0= 1 is one of the roots of the characteristic polynomial.)(b) (5 points) Find a particular solution to the differential equationy000+ y00+ 3y0− 5y = ex+ 1.(c) (1 point) Write down the general solution to the differential equation in part (b).Student Net ID: MATH 286 G1 Page 6 of 125. Let P (t) = [pij(t)] be an n × n matrix of continuous functions (on R) and consider thehomogeneous first order linear systemx0= P (t)x (x(t) ∈ Rn).(a) (3 points) Explain what a fundamental matrix is for this linear system.(b) (3 points) Prove that Φ(t) := expRt0P (s)dsis a fundamental matrix for thislinear system. (Here exp(B) denotes the matrix exponential of a matrix B.)(c) (6 points) Solve the initial value problemx0=0 t20 0x +t0; x(0) =01.Student Net ID: MATH 286 G1 Page 7 of 126. (10 points) Find the general solution to the linear systemx0=1 −4 30 1 60 0 2x.Student Net ID: MATH 286 G1 Page 8 of 127. A 2π-periodic external force of f (t) Newtons is applied to an undamped mass-springsystem with mass m = 3 kg and spring constant k = 27 N/m.(a) (2 points) Write down the equation of motion for the system and determine thenatural frequency of the system.(b) (5 points) Explain why the numerical quantitiesZπ−πf(t) sin(3t)dt andZπ−πf(t) cos(3t)dt,are relevant to the investigation of pure resonance in this system.(c) (5 points) If f(t) = | sin t| for −π < t < π, will pure resonance occur?Student Net ID: MATH 286 G1 Page 9 of 12(d) (6 points) Compute the Fourier series for the function f (t) from part (c) and usethis to find a particular solution xp(t) for the forced system.Student Net ID: MATH 286 G1 Page 10 of 128. Let L > 0 and let f (t) = L − t for 0 < t < L.(a) (5 points) Determine the Fourier sine series for f on the interval [0, L].(b) (5 points) Determine the Fourier cosine series for f on the interval [0, L].Student Net ID: MATH 286 G1 Page 11 of 12(c) (12 points) Consider an L × L square metal plate with vertices (0, 0), (0, L), (L, L)and (L, 0). Suppose that the plate is insulated along its top and bottom edges, thetemperature at the left edge is held at 0 degrees, and the temperature along theright edge is held at u(L, y) = L − y degrees. If u(x, y) denotes the steady-statetemperature distribution in this plate, then the associated boundary-value problemfor u isuxx+ uyy= 0 (0 < x, y < L)uy(x, 0) = uy(x, L) = 0 (0 < x < L)u(0, y) = 0 (0 < y < L)u(L, y) = L − y (0 < y < L).Using the method of separation of variables, find u(x, y).Student Net ID: MATH 286 G1 Page 12 of 12(Extra work