MATH 251Examination I INovember 12, 2007Name:Student Number:Section:This exam has 11 questions for a total of 100 points. In order to obtain full credit for partialcredit problems, all work must be shown. Credit will not be given for an answer notsupported by work. The point value for each question is in parentheses to the right of thequestion number.You may not use a calculator on this exam. Please turn off and put away yourcell phone.1:2:3:4:5:6:7:8:9:10:11:Total:Do not write in this box.MATH 251 EXAMINATION II November 12, 20071. (6 points) Match the sketches of phase portraits for 2x2 homogeneous linear systems withconstant coefficients x′= Ax with the names of their critical points at the origin.I IIIII IVV VI(a) saddle(b) node(c) proper node(d) center(e) spiral(f) improper nodePage 2 of 10MATH 251 EXAMINATION II November 12, 20072. (5 points) When an object with mass 5 kg is attached to a spring, the object stretches thespring by 2 m. A damper with damping coefficient of 4 N/m is attached to the system. Assumethere is no external force acting on the system and that acceleration due to gravity is 10 m/s2.If the object is released 1 m above its equilibrium position and is given an initial downwardvelocity of 3 m/s, which initial value problem describes the displacement of the mass from itsequilibrium position? Take the downward direction to be positive for all displacements andforces.(a) 5y′′+ 4y′+ 25y = 0, y(0) = 1, y′(0) = −3(b) 5y′′+ 4y′+ 25y = 0, y(0) = −1, y′(0) = 3(c) 5y′′+ 4y′+ 10y = 0, y(0) = 1, y′(0) = −3(d) 5y′′+ 4y′+ 10y = 0, y(0) = −1, y′(0) = 33. (5 points) Which of the following functions hasR∞0e−stt2dt as its Laplace transform?(a) t2(b) e−tt2(c) t(d) δ(t − 2)t24. (5 points) If f(t) = 1 − u(t − 1) + 4tu(t − 2) − 10(t3− t)u(t − 5), where u(t − c) = uc(t) is theunit step function, what is f(3)?(a) 1(b) 0(c) 12(d) −212Page 3 of 10MATH 251 EXAMINATION II November 12, 20075. (5 points) Which of the following 2nd order equations is equivalent to the given linear s ystem?x′1= x2x′2= 3x1− 2x2(a) y′′− 3y′+ 2y = 0(b) y′′+ 2y′− 3y = 0(c) y′′− 3y′+ y = 0(d) y′′+ 5y′+ y = 0Page 4 of 10MATH 251 EXAMINATION II November 12, 20076. A spring-mass system is modeled by the initial value problem2y′′+ γy′+ 8y = F (t), γ ≥ 0, y(0) = 3, y′(0) = −4.(a) (6 points) If γ = 0 and F (t) = 0, what is the amplitude of displacement?(b) (3 points) If γ = 0 and F (t) = 3 cos(ωt), for which value(s) of ω will the system undergoresonance?(c) (4 points) If F (t) = 0, for which value(s) of γ will the system not oscillate?Page 5 of 10MATH 251 EXAMINATION II November 12, 20077. (a) (6 points) Compute the following Laplace transform:L{tu(t − 3)}(b) (7 points) Compute the following inverse Laplace transform:L−1ss2+ 4s + 5Page 6 of 10MATH 251 EXAMINATION II November 12, 20078. (14 points) Use Laplace transforms to solve the following initial value problem:y′+ 2y = et−3u(t − 3) + δ(t − 3), y(0) = 3.Recall that u(t − c) = uc(t) is the unit step function.Page 7 of 10MATH 251 EXAMINATION II November 12, 20079. Consider the 2x2 linear homogeneous system with constant coefficients x′= Ax.(a) (4 points) If the eigenvalues of A are ±i, classify the type and stability of the criticalpoint(0, 0).(b) (5 points) If in addition an eigenvector corresponding to the eigenvalue i is2i, thenwrite down the real-valued general solution x(t).Page 8 of 10MATH 251 EXAMINATION II November 12, 200710. Consider the systemx′=3 −41 −2x.(a) (10 points) Find the general solution x(t).(b) (3 points) If x(0) =6βand limt→∞x(t) = 0, what is β?Page 9 of 10MATH 251 EXAMINATION II November 12, 200711. (a) (4 points) Find the critical points of the following nonlinear system:x′= y(1 − x2)y′= x + y(b) (8 points) Linearize the s ystemx′= 1 − yy′= x2− y2around its critical point (1,1) and classify its type and stability.Page 10 of
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