MATH 251Examination IIApril 4, 2011FORM AName:Student Number:Section:This exam has 12 questions for a total of 100 points. In order to obtain full credit forpartial credit problems, all work must be shown. For other problems, points might bededucted, at the sole discretion of the instructor, for an answer not supported by areasonable amount of work. The point value for each question is in parentheses to the right ofthe question number. A table of Laplace transforms is attached as the last page of the exam.Please turn off and put away your cell phone.You may not use a calculator on this exam.1:2thru7:8:9:10:11:12:Total:Do not write in this box.MATH 251 EXAMINATION II April 4, 20111. (12 points) A mass-spring system is described by the equation4u00+ γu0+ ku = F(t).(a) (3 points) Suppose the mass originally stretched the spring 2m to reach its equilibriumposition. What is the spring constant k? (Assume g = 10m/s2to be the gravitationalconstant.)(b) (3 points) Suppose k = 25. For what value(s) of γ would this system be critically damped?(c) (3 points) Suppose γ = 0 and k = 400. What is the natural period of this system?(d) (3 points) True or false: Suppose γ = 10 and k = 10, then every nonzero solution of themass-spring system will cross the equilibrium position more than once. Why or why not?Page 2 of 10MATH 251 EXAMINATION II April 4, 20112. (5 points) Consider the fourth order linear equation2y(4)+ 50y00= 0.What is its general solution?(a) y(t) = C1et+ C2e−t+ C3cos 5t + C4sin 5t(b) y(t) = C1+ C2t + C3cos 5t + C4sin 5t(c) y(t) = C1cos√5t + C2sin√5t + C3t cos√5t + C4t sin√5t(d) y(t) = C1+ C2et+ C3cos 5t + C4sin 5t3. (5 points) Find the Laplace transform L{u2(t) (t2+ 1)}.(a) F (s) = e−2ss2+ 2s3(b) F (s) = e−2s2 − s2s3(c) F (s) = e−2s5s2+ 4s + 2s3(d) F (s) = e−2s5s2− 4s + 2s3Page 3 of 10MATH 251 EXAMINATION II April 4, 20114. (5 points) Evaluate the following definite integralZ∞0e−stδ(t −π6) sin(3t) dt.(a) e−π6s(b) e−π6s3s2+ 9(c) eπ6sss2+ 9(d) −eπ6s5. (5 points) Which system of first order linear equations below is equivalent to the second orderlinear equationy00+ 2y0− y = 2t3?(a)x01= x2x02= 2x1− x2+ 2t3(b)x01= x2x02= x1− 2x2+ 2t3(c)x01= x2x02= −2x1+ x2+ 2t3(d)x01= x2x02= −x1+ 2x2+ 2t3Page 4 of 10MATH 251 EXAMINATION II April 4, 20116. (5 points) Consider a certain system of two first order linear differential equations in twounknowns, x0= Ax, where A is a matrix of real numbers. Suppose one of the eigenvalues ofthe coefficient matrix A is r = 1 + 3i, which has a corresponding eigenvector2 − i−5. Whatis the system’s real-valued general solution?(a) x(t) = C1et2 cos 3t + sin 3t−5 cos 3t+ C2et−cos 3t + 2 sin 3t−5 sin 3t(b) x(t) = C1et2 cos 3t − sin 3t−5 cos 3t+ C2etcos 3t − 2 sin 3t5 sin 3t(c) x(t) = C1et−2 cos 3t − sin 3t5 sin 3t+ C2etcos 3t − 2 sin 3t−5 cos 3t(d) x(t) = C1et−2 cos 3t + sin 3t5 sin 3t+ C2et−cos 3t + 2 sin 3t5 cos 3t7. (5 points) Consider a certain linear system x0= Ax, where A is a matrix of real numbers withdistinct nonzero real eigenvalues. Suppose all of its solutions have a finite limit as t → +∞.Then the critical point (0, 0) must be a(n)(a) (neutrally) stable center.(b) asymptotically stable spiral point.(c) unstable saddle point.(d) asymptotically stable node.Page 5 of 10MATH 251 EXAMINATION II April 4, 20118. (9 points) For each part below, determine whether the statement is true or false. You mustjustify your answers.(a) Consider the linear system x0=0 3−2 0x. The critical point (0, 0) is a (neutrally)stable center.(b) L{(t + 5)2} = L{t + 5}L{t + 5}(c) Suppose f(t) = 2 + u3(t) (t − 1) + u5(t) t2, then f(4) = 5.Page 6 of 10MATH 251 EXAMINATION II April 4, 20119. (14 points) Find the inverse Laplace transform of each function given below.(a) (7 points) F (s) =2s2− 1s3− 3s2− 10s(b) (7 points) F (s) = e−sss2− 10s + 29Page 7 of 10MATH 251 EXAMINATION II April 4, 201110. (14 points) Use the Laplace transform to solve the following initial value problem.y00+ 4y = 2δ(t − 5) − 8u3(t), y(0) = 0, y0(0) = 4.Page 8 of 10MATH 251 EXAMINATION II April 4, 201111. (11 points) Consider the initial value problem.x0=2 1−1 0x, x(0) =−26.(a) (9 points) Solve the initial value problem.(b) (2 points) Classify the type and stability of the critical point at (0, 0).Page 9 of 10MATH 251 EXAMINATION II April 4, 201112. (10 points) Consider the nonlinear system:x0= (2x + y)(2x − y)y0= (x − 2)(y + 2)(a) (3 points) One of the critical points of the system is (1, −2). There are 3 other criticalpoints. Find those other 3 critical points of the system.(b) (7 points) Linearize the system about the point (1, −2). Classify the type and stability ofthe critical point at (1, −2) by examining the linearized system. Be sure to clearly statethe linearized system’s matrix and its eigenvalues.Page 10 of
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