Differential equationsMath 217 — Fall 2009November ExamThis exam contains thirteen problems numbered 1 through 13. Problems 1 – 12 are multiplechoice problems. Problem 13 is a free-response question.Problem 1Two tanks of brine are connected to each other. At time t, tank 1 contains x(t) lbs ofsalt in 100 gal of brine and tank 2 contains y(t) lbs of salt in 200 gal of brine. A brinecontaining a concentrate of .5 lbs/gal flows into tank 1 at a rate of 10 gal/min andthe well mixed brine flows out of tank 2 at a rate of 10 gal/min. Brine from tank 1 ispumped to tank 2 at a rate of 15 gal/min and brine from tank 2 is pumped to tank 1at rate of 5 gal/min. Which of the following differential equations models this system?A) 40x0= y − 6x, 40y0= 6x − 3y.B) x0= y − x, y0= 6x − 3y.C) 40x0= 200 + 6y − 3x, 60y0= 10 − 9y.D) 60x0= 200 + 3y − 10x, 40y0= 300 + x − 3y.E) 40x0= 200 − 6x + y, 40y0= 6x − 3y.F) 40x0= 5 + 3y − 6x, 40y0= 3x − 3y.1Problem 2Write the systemx00= y − 2xy0= x + 2yas a linear differential equation.A) x000− 2x00+ 2x0= 0B) x000+ x00+ x0+ x = 0C) x000− 2x00− 5x = 0D) x000= 0E) x000+ 2x0− 5x = 0F) x000− 2x00+ 2x0− 5x = 02Problem 3It is known that the matrixA =2 1a 2has eigenvalues λ = 1 and λ = 3. What is a?A) −1B) 0C) 1D) 2E) 3F) 43Problem 4Which of the following is a particular solution to the differential equationy00+ y = f (x).A) yp= sin x + cos x +Zf(x) dxB) yp=Zf(x) cos x dx +Zf(x) sin x dxC) yp= − sin xZf(x) sin x dx + cos xZf(x) cos xD) yp= − cos xZf(x) sin x dx + sin xZf(x) cos x dx + 2 sin xE) yp= − sin xZf(x) cos x dx + cos xZf(x) sin x dxF) yp= − cos xZf(x) dx + sin xZf(x) dx4Problem 5LetA =1 −33 7.Which of the following statements about A is correct.I) Any vector [c1c2] (c1and c2are constant) is a generalized eigenvector of A.II) The vector−e4te4t is a solution to the equation x0= Ax.III) The vector−te4tte4t is a solution to the equation x0= Ax.A) Only IB) Only IIC) Only IIID) I and IIE) I and IIIF) II and III5Problem 6Find the general solution tox01= −5x2x02= x1+ 2x2A) x = Ae2tcos(t) − 2 sin(t)− cos(t)+ Be2t2 cos(t) + sin(t)− sin(t)B) x = Ae2t2 cos(t) − sin(t)cos(t)+ Be2tcos(t) + 2 sin(t)sin(t)C) x = Aetcos(2t) + 2 sin(2t)− cos(2t)+ Bet2 cos(2t) + sin(2t)− sin(2t)D) x = Ae2t2 cos(2t) − sin(2t)− cos(t)+ Be2tcos(2t) + 2 sin(2t)− sin(t)E) x = Aet− cos(2t) − 2 sin(2t)cos(2t)+ Bet2 cos(2t) − sin(2t)sin(2t)F) x = Aetcos(2t) − 2 sin(2t)− cos(2t)+ Bet2 cos(2t) + sin(2t)− sin(2t)6Problem 7Which of the following is the general solution to the systemx0=0 −22 −4x?A) x = c1−2−2e−2t+ c201te−2tB) x = c111te−2t+ c201e−2tC) x = c122e−2t+ c210t +22e−2tD) x = c122e2t+ c222t +10e2tE) x = c1−2−2e−2t+ c2−2−2t +10e−2tF) x = c1−2−2e−2t+ c2−2−2t +01e−2t7Problem 8What is the form of a particular solution ofx0=4 −32 −3x +154e−2t+12cos 2t?Below, a, b, c, and d denote vector coefficients.A) ae−2t+ b cos 2tB) ate−2t+ b cos 2tC) ae−2t+ b cos 2t + c sin 2tD) ate−2t+ b cos 2t + c sin 2tE) ae−2t+ bte−2t+ c cos 2t + d sin 2tF) ae−2t+ bte−2t+ ct cos 2t + dt sin 2t8Problem 9Solve the initial value problemx0=3 −15 −3x, x(0) =5−3.A)x =7e2t− 2e−2t7e2t− 10e−2tB) x =35e2t− 2e−2t7e2t− 10e−2tC) x =7e2t− 2e−2t7e−2t− 10e2tD) x =7e2t− 10e−2t7e2t− 2e−2tE) x =7e−2t− 2e2t7e−2t− 10e2tF) x =7e−2t− 10e2t7e−2t− 2e2t9Problem 10Suppose A is a 2×2 matrix with eigenvalues λ1= 2, λ2= −2 and associated eigen-vectors [1 1]Tand [−3 1]Trespectively. Which of the following is a particular solutiontox0= Ax + 4ete−t.A) xp=e2t− 3e−2te2t+ e−2tB) xp=4et43e−tC) xp=−e−t− e−3t−13e3t+ etD) xp=−13e3t+ et−e−t− e−3tE) xp=−4et−43e−tF) xp=−4e−t−43et10Problem 11A mass-spring-dashpost system with m = 1, c = 4, and k = 5 is acted upon by theexternal for FE(t) = 10 cos 3t. The mass starts in its equilibrium position at rest. Whatis the steady period solution xsp?A) xsp=14(cos(3t) − 3 sin(3t))B) xsp=14(− cos(3t) + 3 sin(3t))C) xsp=14(cos(t) − 7 sin(t))D) xsp=14e−2t(cos(t) − 7 sin(t))E) xsp=14(− cos(t) + 7 sin(t))F) xsp=14(cos(3t) − 7 sin(3t))11Problem 12Which of the following differential equations describes a spring-mass system whichexhibits the phenomenon of resonance?A) x00+ 3x0+ 2x = 0B) x00+ 3x0+ 2x = cos 2tC) x00+ 3x0+ 2x = sin tD) x00+ x =12sin tE) x00+ x = 0F) x00+ 2x = cos 2t12Math 217 Differential Equation, November 17, 2009Name: Student-ID: Section: 9 – 10 Tang 11 – 12 MoenThe following problem is a free-response question. You should justify your answers.Problem 13a) Find the general solution ofx0=3 4 50 5 40 0 3x.b) Solve the initial value problemx0=3 4 50 5 40 0 3x, x(0)
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