MATH 251FINAL EXAMINATIONDecember 17, 2008Name:Student Number:Section:This exam has 17 questions for a total of 60 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. A list of Laplace transforms is attached as the last page of this booklet. It canbe removed for easy reference during the examination.You may not use a calculator on this exam. Please turn off and put away yourcell phone.1-12:13:14:15:16:17:Total:Do not write in this box.MATH 251 FINAL EXAMINATION December 17, 20081. B2. A3. C4. C5. A6. C7. B8. D9. B10. B11. D12. D13. (a) True(b) True(c) False(d) True(e) False(f) TruePage 2 of 6MATH 251 FINAL EXAMINATION December 17, 200814. (14 points) In each part below, consider a certain system of two first order linear differentialequations in two unknowns, x0= Ax.(a) (4 points) Suppose that the system’s general solution isx(t) = C132e−t+ C2−12e−6t.Classify the type and stability of the system’s critical point at (0, 0).NODE, ASYMPTOTICALLY STABLE(b) (4 points) Suppose the only eigenvalue of the coefficient matrix A is 2, which has corre-sponding eigenvectors10and01. Write down the system’s general solution.y = c1e2t10+ c2e2t01(c) (3 points) Classify the type and stability of the critical point at (0, 0) for the systemdescribed in (b).PROPER NODE, UNSTABLE(d) (3 points) Suppose A has eigenvalues 9i and −9i, classify the type and stability of thesystem’s critical point at (0, 0).CENTER, STABLEPage 3 of 6MATH 251 FINAL EXAMINATION December 17, 200815. (14 points) Find all positive eigenvalues and corresponding eigenfunctions of the boundaryvalue problemX00+ λX = 0, X(0) = 0, X0(4) = 0.General Solution: X = c1cos√λx + c2sin√λx (2pt)X0= −c1√λ sin√λx + c2√λ cos√λx (2pt)X(0) = c1= 0 (1pt)X0(4) = −c1√λ sin 4√λ + c2√λ cos 4√λ = 0 (1pt)c1= 0 ⇒ c26= 0, λ 6= 0 ⇒ cos 4√λ = 0 (2pt)4√λ = nπ/2, n odd (2pt)Eigenvalues: λ = (nπ/8)2, n odd (2pt)Eigenfunctions: X =sinnπx8, n odd (2pt)Page 4 of 6MATH251 FINAL EXAMINATION December 17, 200816. (16points) Let f(x) = x3, 0 < x< 1.(a) (4 points) Consider the odd periodic extension, of period T = 2, of f (x). Sketch 3 periods,on the interval −3 < x < 3, of this odd periodic extension.(b) (2 points) Find a10, the10th cosine coefficient of the Fourier series of the odd periodicextension in (a).a10= 0(c) (6 points) Which of the integrals below can be used to find the Fourier sine coefficients ofthe odd periodic extension in (a)?(i) bn=12Z10x3sinnπx2dx(ii) bn=Z1−1x3sinnπx2dx(iii) bn= 2Z10x3sin(nπx) dx(iv) bn=Z1−1x3cos(nπx) dx(d)(4 points) To what value does the Fourier series converge at x = −1? At x =12?0; 1/8Page5 of 6MATH 251 FINAL EXAMINATION December 17, 200817. (16 points) Suppose the temperature distribution function u(x, t) of a rod that has both endsperfectly insulated is given by the initial-boundary value problem9uxx= ut, 0 < x < 4, t > 0ux(0, t) = 0, ux(4, t) = 0,u(x, 0) = 2 − cos(πx) − 7 cos(5πx).(a) (14 points) Find the particular solution of the above initial-boundary value problem.Assume u(x, t) = X(x)T (t)9X00T = XT0X00X=T09T= −λX00+ λX = 0, T0+ 9λT = 0 (2pt)ux(0, t) = 0, ux(4, t) = 0 ⇒ X0(0)T (t) = 0, X0(4)T (t) = 0 ⇒ X0(0) = 0, X0(4) = 0 (2pt)λn= (nπ/4)2(2pt)Xn= cos(nπx/4) (2pt)Tn= cne−9(nπ/4)2t(2pt)u(x, t) = c0/2 +P∞n=1cne−9(nπ/4)2tcos(nπx/4) (1pt)u(x, 0) = 2 − cos(πx) − 7 cos(5πx) = c0/2 +P∞n=1cncos(nπx/4)c0= 4, c4= −1, c20= −7; otherwise cn= 0 (2pt)Solution: u(x, t) = 2 − e−9π2tcos(πx) − 7e−9(5π)2tcos(5πx) (1pt)(b) (2 points) What is limt→∞u(3, t)?lim = 2Page 6 of
View Full Document
Unlocking...