Differential equationsMath 217 — Fall 2009Final Exam DecemberThis final exam contains twenty problems numbered 1 through 20. All problems are multiplechoice problems. Every problem counts 5 points.Problem 1Let x(t) be the solution to the initial value problemtx00+ x0+ tx = 0, x(0) = 1, x0(0) = 0.What equation does X(s) = L{x}(s) satisfy?A) (s + 1)X0+ X = 0 B) (s2+ 1)X0+ X = 0C) (s + 1)X0+ sX = 0 D) (s2+ 1)X0+ sX = 0E) (s + 1)X0+ s2X = 0 E) (s2+ 1)X00+ sX0= 0.1Problem 2Let x(t) be the solution ofx0=1 −11 3x, x(0) =32.What is x(1).A)32e2B)6−1e2C)−27e2D)11e2E)31e2F)−36e22Problem 3Which of the following differential equations is exact?A) y3+ 3x2y y0= 0 B) 3xyexydx + x3exydy = 0C) y2dx + x2dy = 0 D) xy + xydydx= 0E) yexydx + xexydy = 0 F) (x2y2+ x) dx + (x2+ y2) dy = 03Problem 4Find the general solution to the following initial value problemx0=4 23 −1x.A)c1e−2t+ c2e5tc1e−2t+ c2e5tB)c1e−2t+ 2c2e5tc1e−2t+ c2e5tC)c1e−2t+ 2c2e5t−c1e−2t+ c2e5tD)c1e−2t+ 2c2e5t−3c1e−2t+ c2e5tE)c1e−2t+ c2e5t−3c1e−2t+ 2c2e5tF)c1e−2t+ c2e5t−3c1e−2t− 2c2e5t4Problem 5Find the inverse Laplace transform ofF (s) =1√s − 1.Recall Γ(1/2) =√π.A) etqtπB)1√πtC) etpπtD)qtπE)et√πtF)√πt5Problem 6Consider the problemx0=1 23 4x.Which of the following statements are true?I) The matrix has a defect eigenvalue.II) The problem has two linearly independent solutions.III) The problem has a nonzero solution that satisfies x(0) = [00].A) Only I B) Only II C) Only IIID) I and II E) I and III F) II and III6Problem 7What value of m will yield resonance in the following mass-spring system?mx00+ x = 10 cos(3t)A) m = 3 B) m = 1/3 C) m = 1/9D) m = 1/√3 E) m = 9 F) m = 17Problem 8Suppose yc= c1x + c2x2is the complementary solution toy00−2xy0+2x2y =√x.Find a particular solutionA)43x5/2B) −43x5/2C) −23x3/2D)12x1/2E) −23x3/2F)43x3/28Problem 9Find the general solution ofy0= 1 + 2x + y + 2xy.A) ln |1 + 2y| = x + C B) ln |1 + y| = x − x2+ CC) ln |1 + y| = x + x2+ C D) ln |1 + y| = x − x2+ CE) ln |1 + 2y| = x + x2+ C F) ln |1 + 2y| =12x + C9Problem 10Let F (s) and G(s) be two functions such thatL−1{F (s)} = sin 2t and L−1{G(s)} = et.What is L−1F (s)G(s)?A)25et−15sin 2t −15cos 2t B)25et−15sin 2t −25cos 2tC)25et−15sin 2t +15cos 2t D)25et−15sin 2t +25cos 2tE)25et+15sin 2t −15cos 2t F)25et+15sin 2t −25cos 2t10Problem 11Find the solution of the equationx00− x0− 6x = f(t), x(0) = 2, x0(0) = −1.A)75e−2t+35e3t+ f(t) B)15e−2tf(t) +15e3tf(t)C)15Rt0(e−2τ+ e3τ)f(t − τ )dτ D)Rt0(75e−2τ+35e3τ)f(t − τ )dτE)75e−2t+35e3t+15Rt0(e−2τ+ e3τ)f(t − τ )dτ F)Rt0f(t − τ )dτ11Problem 12Calculate the Laplace transformLet− 1t.A) lns−1sB) lnss−1C)es−1sD)1s2(s−1)E)1s−1−1sF)1s−112Problem 13Two 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 100 gal of brine. Brinefrom tank 1 flows into tank 2 at 10 gal/min and brine from tank 2 flows into tank 1same rate. Which of the following systems models this situation.A) 10x0= y −x, 10y0= x − y B) 10x0= x − y, 10y0= y −xC) x0= 10y −10x, y0= 10x − 10y D) 10x0= y, 10y0= xE) 10x0= x + y 10y0= x − y F) x0= 10x − 10y, y0= 10y −10x13Problem 14Consider the following Logistic population modeldPdt= 2MP −P2.What is the equilibrium population?A) 2M B) M/2 C) MD) 2 E) 4M F) M/414Problem 15Solvedydx− 2y = 3e2x, y(0) = 0.What is y(1)?A) −3e2B) −2e2C) −e2D) e2E) 2e2F) 3e215Problem 16Compute L{te−tsin t}.A)2(s+1)(s2+1)2B)2(s+1)(s2+2s+2)2C)s+1(s2+2s+2)2D)2s(s2+2s+2)2E)(s+1)(s2+1)2F)s(s2+1)216Problem 17Solvey00+ 2y0+ y = 0, y(0) = 5, y0(0) = −3.What is y(1)?A)3eB)5eC)7eD)9eE)11eF)13e17Problem 18Let y =P∞n=0cnxnbe a nonzero solution of the equation (x − 3)y0+ 2y = 0. What isthe radius of convergence of y(x)?A) 0 B) 1 C) 2 D) 3 E) 4 F) ∞18Problem 19Suppose y =P∞n=1cnxnis a solution to(x − 1) y00+ (x − 2) y0+ y = 0.What is the recurrence relation between the coefficients?A) cn+2=(n+1)cn+1+cn(n+2)B) cn+2=(n+2)cn+1+(n+1)cn(n+2)C) cn+2=(n−2)cn+1+cn(n+1)D) cn+2=(n−2)cn+1+(n+1)cn(n+2)E) cn+2=(n+1)cn+1+ncn(n+2)F) cn+2=(n−2)cn+1+cn(n+2)19Problem 20Solve the initial value problemx00+ 4x = sin t, x(0) = 1, x0(0) = −1A) x =13sin t B) x = cos(2t) +13sin(2t) +13sin tC) x = e−2t−23e2t+13sin t D) x = cos(2t) +13sin(2t) −23sin tE) x = cos(2t) −23sin(2t) +13sin t F) x =16e−2t+56e2t+13sin
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