MATH 251Final ExaminationDecember 15, 2010FORM AName:Student Number:Section:This exam has 18 questions for a total of 150 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.1thru14:15:16:17:18:Total:Do not write in this box.MATH 251 FINAL EXAMINATION December 15, 20101. (6 points) Find the solution of the initial value problemy0=cos tsin y, y(0) =π2.(a) y = cos−1(− sin t)(b) y = cos−1(sin t − 1)(c) y = − sin−1(cos t)(d) y = sin−1(− cos t +π2)2. (6 points) Which initial or boundary value problem below is guaranteed to have a uniquesolution according to the Existence and Uniqueness theorems?(a) y00+ sin(5t)y0− cos(10t)y = π, y(0) = 1, y0(0) = −1.(b) (t + 2)y0− e−ty = t, y(−2) = 0.(c) y00+ 100y = 0, y(0) = 9, y(2π) = −10.(d) t2y00+ ty0+ y = 0, y(0) = −2, y0(0) = 3.Page 2 of 12MATH 251 FINAL EXAMINATION December 15, 20103. (6 points) Let y1(t) and y2(t) be any two solutions of the second order linear equation2ty00+ 4y0− t3cot(2t)y = 0.What is the general form of their Wronskian, W (y1, y2)(t)?(a) Ct4(b) Ce−4t(c) Ce2t(d) Ct−24. (6 points) Which of the functions below is a particular solution of the nonhomogeneous linearequationy00+ 3y0+ 2y = 2t + 1?(a) y = 2e−t(b) y = t − 1(c) y = −11e−2t(d) y = 4e−t+ e−2t+ 2t + 1Page 3 of 12MATH 251 FINAL EXAMINATION December 15, 20105. (6 points) A certain mass-spring system is described by the equation3u00+ γu0+ 12u = 0.Find all values of γ such that the system would be underdamped.(a) −12 < γ < 12(b) 0 < γ < 12(c) γ ≤ 144(d) 0 ≤ γ < 1446. (6 points) Find the general solution of the fourth order linear equationy(4)+ 3y(3)− 4y00= 0.(a) y(t) = C1cos t + C2sin t + C3et+ C4e−4t(b) y(t) = C1t + C2t2+ C3e−t+ C4e4t(c) y(t) = C1+ C2t + C3et+ C4e−4t(d) y(t) = C1e−t+ C2e4t+ C3te−t+ C4te4tPage 4 of 12MATH 251 FINAL EXAMINATION December 15, 20107. (6 points) The inverse Laplace transform of F (s) = e−4ss − 1s2+ 7s + 10is(a) f(t) = u4(t)(e−2t+8− 2e−5t+20)(b) f(t) = u4(t)(e−2t−8− 2e−5t−20)(c) f(t) = u4(t)(−e−2t−8+ 2e−5t−20)(d) f(t) = u4(t)(−e−2t+8+ 2e−5t+20)8. (6 points) Consider the linear system below.x0=8 00 8x.Which statement below is true?(a) The critical point (0, 0) is an asymptotically stable proper node.(b) The critical point (0, 0) is an unstable improper node.(c) x = C1e8t10+ C2e8t0−1is a general solution of the system.(d) x = C1e−8t11+ C2e−8ttt + 1is a general solution of the system.Page 5 of 12MATH 251 FINAL EXAMINATION December 15, 20109. (6 points) Given that the point (2, −2) is a critical point of the nonlinear system of equationsx0= x2− y2y0= xy + 2x − y − 2.This critical point (2, −2) is an(a) unstable node.(b) unstable saddle point.(c) asymptotically stable improper node.(d) asymptotically stable spiral point.10. (6 points) Consider the two linear partial differential equations.(I) uxx= 2ut+ 1(II) uxx= 2ut+ 3uUse the substitution u(x, t) = X(x)T (t) and attempt to separate each equation into two ordi-nary differential equations. Which statement below is true?(a) Neither equation is separable.(b) Only (I) is separable.(c) Only (II) is separable.(d) Both equations are separable.Page 6 of 12MATH 251 FINAL EXAMINATION December 15, 201011. (6 points) Find the Fourier cosine coefficient corresponding to n = 2, a2, of the Fourier seriesof period T = 2π representing the function f (x) = 6 cos 2x.(a) a2= 0(b) a2= 6(c) a2=3π(d) a2=−1π12. (6 points) Consider the Fourier series (of period 20) representingf(x) = x5, −10 < x < 10, f(x + 20) = f(x).Which statement below is true?(a) The Fourier series is a cosine series.(b) The Fourier series is a sine series.(c) The Fourier series is neither a cosine series nor a sine series.(d) The function does not have a Fourier series because it is not periodic.Page 7 of 12MATH 251 FINAL EXAMINATION December 15, 201013. (6 points) Find the steady-state solution, v(x), of the heat conduction problem with nonho-mogeneous boundary conditions:α2uxx= ut, 0 < x < 5, t > 0u(0, t) + ux(0, t) = 0, u(5, t) = 8,u(x, 0) = f(x) = 10x + 8.(a) v(x) =85x(b) v(x) = 10x + 8(c) v(x) = −x + 13(d) v(x) = 2x − 214. (6 points) Which function below is a solution of the wave equation boundary value problem4uxx= utt,u(0, t) = 0, u(3, t) = 0?(a) u(x, t) = 5 cos(4πt) sin(2πx)(b) u(x, t) = t − sin(9πx)(c) u(x, t) = e−4π2tcos(πx)(d) u(x, t) = 6e4xtsin(3πx)Page 8 of 12MATH 251 FINAL EXAMINATION December 15, 201015. (14 points) Use the Laplace transform to solve the following initial value problem.y00+ 36y = 3δ(t − 1), y(0) = −4, y0(0) = 0.No credit will be given if the Laplace transform is not used to solve this problem.Page 9 of 12MATH 251 FINAL EXAMINATION December 15, 201016. (16 points) Consider the two-point boundary value problemX00+ λX = 0, X0(0) = 0, X(π) = 0.Find all positive eigenvalues and corresponding eigenfunctions of the boundary value problem.Page 10 of 12MATH 251 FINAL EXAMINATION December 15, 201017. (18 points) Let f (x) = 4, 0 < x < 2.(a) (4 points) Consider the odd periodic extension, of period T = 4, of f(x). Sketch 3 periods,on the interval −6 < x < 6, of this odd periodic extension.(b) (5 points) Find a10, the tenth cosine coefficient of the Fourier series of the periodic functiondescribed in (a).(c) (5 points) Which of the integrals below can be used to find the Fourier sine coefficients ofthe odd periodic extension in (a)?(i) bn=12Z20sinnπx4dx(ii) bn= 2Z2−2sinnπx4dx(iii) bn= 2Z20sinnπx2dx(iv) bn= 4Z20sinnπx2dx(d) (4 points) To what value does the Fourier series of this odd periodic extension convergeat x = −2? At x = 5?Page 11 of 12MATH 251 FINAL EXAMINATION December 15, 201018. (18 points) Suppose the temperature distribution function u(x, t) of a rod that has both endskept at different temperatures is given by the initial-boundary value problem4uxx= ut, 0 < x < 1, t > 0u(0, t) = 60, u(1, t) = 40,u(x, 0) = 60 − 20x − 30 sin(2πx) + 50 sin(7πx).(a) (4 points) Find the steady-state solution of the
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