1(!l. 6(t - c) e~ F(s) 1. sin at. cosat. sinh at. cosh at. (s=~'2+b'2 8-a. uc(t) uc(t)./~_~)

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1 Differential Equations (Math 27) Final Exam December 8, 23 This exam contains twenty multiple-choice problems worth five points each for an exam total of points. For each problem, mark your answer card with the letter corresponding to the only correct answer. You may refer to the following table during the exam. Table of Laolace Transforms (!l F(s).! --'- eat -L.-0. tn sin at cosat sinh at cosh at eat sin bt egtcosbt n! -~ ~ ~ ~ ~ a a & (s=~'2+b'2 8-a (~~_a)2+b2 treat n! (s-a)n+l uc(t) uc(t)./~_~) e-c.s -'- e~f(8) 6(t - c) e~

2 . Find the general solution of the following differential equation, and determine how the solutions behave as t -+. tv + 2] 6, t>o (A)., -+ - (B)., (C)., (D)., (E) Y -+ 0 (F) y-+2 (G) Y -+ 3 (H),,-+6 (I) () -+ The limit cannot be determined because it depends on the initial conditions. 2. Consider the initial value problem (t2-3t)v' + tv - (t + 3)y = 0, y(l) = 2, V(l) =. The existence and uniqueness theorem for linear differential equations guarantees that there will be a unique solution. Determine the longest interval in which this solution is certain to exist. -oo<t<o -<t<2 -oo<t<3 (D) - < t < (E) O<t<2 (F) O<t<3 (0) 0 < t < 2<t<3 (I) 2<t< () 3<t<oo

3 3. Find the general solution of the following differential equation. (A) = Cl est + C2e3t (8) = clest + C2t3e3 (C) = Cl e-3t + C2e-3t (D) = Cl e-st + C2te-3t (E) = Cl est + C2e-St (F) = Clt3 + C2t3 (G) Y = Cl t' + C2t'ln t (H) Y = Cl t-3 + C2t-3 (I) (J) y = Cl t-3 + C2t-3ln t Y = Clt3 + C2t-3 '/' +6J! +9y = 0 4. Consider a spring system in which a 6 pound object stretches the spring six inches. (There is no damping.) What external force would cause resonance to occur? (A) F(t) = t (8) F(t) = et (C) F(t) = te' (D) F(t) = est (E) F(t) = e-8t (F) F(t) = 2cos(.[2t) (G) F(t) = sin( -:j;t) (H) F(t) = 5cos(lt) (I) F(t) = 6sin( ~t) (J) F(t) = 3cos(8t) - sin(8t)

4 5. Consider the differential equation 2x(x - 5)2/' + xi' + (x - 5), = 0 xo =3. Classify the point (A) It is an ordinary point of the differential equation. (B) It is a regular singular point of the differential equation. (C) It is an irregular singular point of the differential equation. 6. Consider the differential equation 2x(x - 5)2/' + xi' + (x - 5)y = o. Determine a lower bound for the radius of convergence of the series solutions about Xo = 3 for this differential equation. (A) 0 (B) (C) 2 (D) 3 (E) 4 (F)5 (G) 6 (H)7 (I) 8 (J)

5 7. Consider the differential equation XV' - 4xy' + 4cr O. Note that Xo = 0 is a regular singular point. Find the fond of two linearly independent series solutions about Xo = O. (Hint: In the book, there are some fonds in which "bn" is used for the coefficients in the second solution and some forms in which "Cn" is used. However, it is not important which variable is used. For the sake of consistency, "bn" is used in each of the answers below.) (A) Y = E anxn (B) Y = Eanxn (C) (D) Y = E anxn Y = X E anxn (E) = X Eanxn (F) (G) (H) (I) (J) Y = X E anxn Y = X4 E anxn tt=o Y = X4 E anxn Y = x2 E anxn Y = x2 E anx" /2 = Ebnxn Y2 = YllnX + Ebftxn n=l 2 = IlnX + x Eb"x",,=0 /2 = ~ b"x",,=0 Y2 = ayiinx + Eb"x" R=O Y2 = YIInX + X' Eb"x",,=0 Y2 = X E b"x" /2 = a/ln x + x E bnxn 8=0 '2 = Z2 E bnzn 'Y2 = 'YlinX +:r? Ebnxn n=

6 8. Consider the differential equation X2y" + Xy' + xy = O. It turns out that there is a series solution of the form y = E anxn. (You do not need to verify this.) Set~:i ~, and find al. 40 = J (A) - (8) -2 (C) - "3 (D) -4 (E) -"6 (F) -9 (G) -2 (H) -IS (I) -"36 (J) 0

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9 3. Each of the following is the general solution of a system of differential equations. Consider the phase portrait for each. For which one(s) is the origin a node? (I) X=Cl =Cl (lli) :J: = C (A) L ll, and ill (B) I and II only (C) I and ill only (D) II and ill only (E) I only (F) n only (G) ill only (H) none of these 4. Consider the following matrix. A= r e K r 3e ( A ~ $ J,L lilt e..} u.:j ;.L i ~ "~I.J. ) Suppose S is a maximal set of linearly independent eigenvectors for A. (In other words, S contains as many eigenvectors as possible.) How many vectors must S contain? (Hint: This problem requires understanding, not computation.) (A) exactly (B) exactly 2 (C) exactly 3 (D) exactly 4 (E) exactly 5 (F) The number of elements in S could be anything from to 5, depending on the value of K.

10 5, Consider the following system of differential equations. x= 3 ~)x The eigenvalues and corresponding eigenvectors for the matrix are as follows. ), +3i A2 3i e) ~(2) \;) (~i) (You do not need to verify this.) Find the general solution of this system, expressed in tends of real-valued functions. x (B) x = Cle3t (D) x X=Cl~ cost -sin t cost cost + C2e3t sint -sin t (E) x Cl et (C~S 3t sm3t + ~e t ( sin 3t -cos 3t (F) x t( COS 3t ) t ( sin 3t Cl e -sin 3t + C2e cos 3t ) (G) x = clet (H) x = Clef CDS 3t CDS 3t cos 3t -cos 3t + C2et ( si~ 3t -sm 3t + t sin 3t sin 3t

11 6. Consider the following nonhomogeneousystem of differential equations. x' = (~ - -2I+) 0-2t-3 The general solution of the corresponding homogeneousystem is as follows. ~ = Cl(;) +C2(2t~ ) (You do not need to verify this.) Find v( t), a particular solution of the nonhomogeneous system. (Hint: Due to the form of g( t) = (- 2~-3 ), the method of undetemined coefficients cannot be used. In addition, the coefficient matrix A = ~ :=; is not diagonalizable, so the method of diagonalization cannot be used. Therefore the method of variation of parameters must be used.) (C) v(t) = v(t) = (E) v(t) = (F) v(t) = 0 t-2 + 2t-3 t- + lnt 0 t- + lnt lnt 2t- -t-2 (H) v(t) = -2t-2 2t-3 (I) (J)

12 7. Find all solutions of the following boundary value problem. (A) = 0 (B) = (C) = cos2x (D) = sin 2x (E) = cos 2x + sin 2x (F), = Cl COS 2x, where Cl is arbitrary (G) = c2sin2x, where C2 is arbitrary /' + 4y = 0,.,(0) = I, Y(r) = (II) = Cl COS 2x + sin 2x, where Cl is arbitrary (I) = cos2x + c2sin2x, where C2 is arbitrary (J) no solution 8. Exactly two of the following statements are false. Which two are they? (I) If g( z) is an even function, then it is symmetric with respecto the y-axis. (ll) /(x) = X3 + cosx is an odd function. (Ill) /(z) = Z3. cosx is an odd function. (IV) If g(z) is an even function, then J~L g(x)dx = O.

13 br- Compute the coefficients

14 20. Solve the following heat conduction partial differential equation. 9uxx = Ut, u(o, t) = 0, u(3, t) = 0, u(x,o) = sin27rx -~ 2~ (H) U(X,t) = Ee-n (I) n= 'IriSin T U(X,t) = Ee-3n2rtsin27rX. () u(x,t) = Ee-3ft2w2tsin6rx -=