N e t I D : N a m e :Midterm#3MATH 286—Differential Equations PlusThursday, April 17• No notes, p ersonal aids or calculator s are permitted.• Answer all questions in the space provided. If you require more space to write your answer, you may continueon the back of the page. There is a blank page at the end of the exam for rough work.• Explain your work! Little or no points will be given for a correct answer with no explanation of h ow you got it.Good luck!Problem 1. (10 poi nts) Let A be a 5 × 5 matrix with eigenvalues ±3i, 1, 1, 1.(a) Suppose that the eigenvalue λ = 1 has defect 1. Does the equation x′= Ax have (nonzero) so lutions of one ofthe following forms?(v1t + v2) etv1t22+ v2t + v3etv1t36+ v2t22+ v3t + v4et(v1t + v2) sin (3t) v1etcos(3t)Circle those that are solutions (for appropriate choices of the coefficients v1, v2, v3, v4).(b) Now, consider the differential equation x′= Ax +3t2, 0, co s (t), 0, −1T. Write down a particular solutionxpwith undetermined coefficients.Problem 2. (10 points) Three brine tanks T1, T2, T3are connected as indicated in the sketch below.The mixtur es in each tank are kept uniform by stirring.Suppose that the mixture circulates between the tank s atthe rate of 10gal/min. T1and T3contain 100gal of brineand T2contains 50gal .Denote by xi(t) the amount (in pounds) o f salt in tank Tiat time t (in minutes). D erive a system of linear differentialequations for the xi.(Do not solve the system.)T1100galT250galT3100gal10gal/minMichael Brannan Armin [email protected] [email protected]/4Problem 3. (20 points) Let A =1 −44 −7.(a) Find two linearly independent s olutions to the linear system x′(t) = Ax(t).(b) Compute etA.Michael Brannan Armin [email protected] [email protected]/4Problem 4. (20 points) Let A be a 3 × 3 mat rix such that etA=1 + t −t −t − t2t 1 − t t − t20 0 1.(a) What are the eigenvalues of A and what are their defects?(b) Solve the initial value problem x′(t) = Ax(t), x(0) =001.(c) Find a particular solution to the inhomogeneous linear system x′(t) = Ax(t) +02/t30.(d) Find the matrix A.Michael Brannan Armin [email protected] [email protected]/4Problem 5. (15 points) Find four independent real-valued solutions ofx′=3 −4 1 04 3 0 10 0 3 −40 0 4 3x.You may use that the characteristic poly nomial has the repeated roots 3 ± 4i. Moreover, you may use thatv2= (0 0 1 i)Tis a generalized eigenvector of rank 2 for λ = 3 − 4i.Michael Brannan Armin [email protected]
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