3! + 4! + Binomial series: if α is a nonnegative integer, the series terminates. Otherwise, the series converges if x < 1 but diverges if x > 1.

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1 Page 1 Name: ID: Section: This exam has 16 questions: 14 multiple choice questions worth 5 points each. hand graded questions worth 15 points each. Important: No graphing calculators! Any non-graphing scientific calculator is fine. For the multiple choice questions, mark your answer on the answer card. Show all your work for the written problems. You will be graded on the ease of reading your solution as well as for your work. You are allowed both sides of 3 5 note cheat card for the exam. Power Series Representations The following power series converge to the indicated functions for all x. e x x n = n! = 1 + x + x! + x3 3! + cos x = sin x = cosh x = sinh x = ln(1 + x) = n=0 ( 1) n x n n=0 n=0 n=0 n=0 (n)! ( 1) n x n+1 (n + 1)! = 1 x! + x4 4! + = x x3 3! + x5 5! x n (n)! = 1 + x! + x4 4! + x n+1 (n + 1)! = x + x3 3! + x5 5! + ( 1) n+1 x n n=1 n = x x + x3 3 Geometric series: converges if x < 1 but diverges if x > x = x n = 1 + x + x + x 3 + n=0 Binomial series: if α is a nonnegative integer, the series terminates. Otherwise, the series converges if x < 1 but diverges if x > 1. (1 + x) α = 1 + αx + α(α 1) x +! α(α 1)(α ) x 3 + 3!

2 Page 1. Simple Harmonic Motion: Mechanical Vibrations Determine the period and frequency of the simple harmonic motion of a body of mass 0.75 kg on the end of a spring with spring constant 48 N/m. a. frequency = π 3 s, period= 3 π Hz b. frequency = π 3 Hz, period= 3 π s c. frequency = 3 π Hz, period= π 3 s d. frequency = 4 π Hz, period= π 4 s e. frequency = 4 π s, period= π 4 Hz f. frequency = π 4 Hz, period= 4 π s g. frequency = 8 π Hz, period= π 8 s h. None of the above Solution. Ans.: d. k We are given m = 0.75 kg and k = 48 N/m. The circular frequency is ω 0 = rad/s. The period is T = π ω 0 = π = π 8 4 period (in Hz), so ν = 1 = 4 Hz. T π = 48 = 8 m 0.75 s. The frequency ν is equal to the reciprocal of the

3 Page 3. Simple Harmonic Motion: Mechanical Vibrations A body with mass m = 1 kg is attached to the end of a spring that is stretched m by a force of 100N. It is set in motion with initial position x 0 = 1 m and initial velocity v 0 = 5 m/s. Find the position function of the body. a. x(t) 5 cos(10t ) b. x(t) 5 cos(10t ) c. x(t) 1 5 cos(10t ) d. x(t) 1 5 cos(10t ) e. x(t) 1 5 cos(10t ) f. None of the above. Solution. Ans.: e. This is Example 1, page 139. Note that 0 < α < π, so (c) and (d) cannot be correct.

4 Page 4 3. Simple Harmonic Motion: Mechanical Vibrations A mass m = 4 kg is attached to both a spring (with spring constant k = 169) and a dashpot (with damping constant c = 0). The mass is set in motion with initial position x 0 = 4 m and initial velocity v 0 = 16 m/s. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. a. Underdamped. x(t) e 5t/ cos(6t 0.854) b. Critically damped. x(t) e 5t/ cos(6t 0.854) c. Overdamped. x(t) e 5t/ cos(6t 0.854) d. Underdamped. x(t) e 5t/ sin(6t 0.854) e. Underdamped. x(t) e 5t/ sin(6t ) f. None of the above. Solution. Ans.: a. This is.4 #19

5 Page 5 4. Simple Harmonic Motion: Simple Pendulum Two pendulums are of lengths L 1 and L and when located at the respective distances R 1 and R from the center of the earth have periods p 1 and p, respectively. Which of the following statements is true? a. b. c. d. p 1 p = R 1 L1 R L p 1 p = R L1 R 1 L p 1 p = R 1 L R L1 p 1 p = R L R 1 L1 e. All the above f. None of the above Solution. Ans.: a. This is.4 #9.

6 Page 6 5. Simple Harmonic Motion: Simple Pendulum Most grandfather clocks have pendulums with adjustable lengths. One such clock loses 10 minutes per day when the length of its pendulum is 30 inches. With what length pendulum will this clock keep perfect time? a inches b inches c inches d inches e inches f inches Solution. Ans: d..4 #8.

7 Page 7 6. Elementary Power Series The power series 1 + x + x + x3 + represents which of the following functions? 4 8 a. cosh x b. sinh x c. x d. 1 1 x e. (1 + x) 1/ f. None of the above Solution. Ans.: c. This is a geometric series: 1 + x + x 4 + x3 8 + = 1 + x ( x ) ( x ) = 1 x = x.

8 Page 8 7. Power Series Operations: Multiplication The power series x 1 8 x x3 + represents which of the following functions? a. cosh x b. sinh x c. x d. 1 1 x e. (1 + x) 1/ f. None of the above Solution. Ans.: e. Note: (a) only involves even powers, (b) only odd powers. Choice (c) is the answer to the last question. By elimination, only (d) and (e) are left and a check shows that (e) is correct.

9 Page 9 8. Analytic functions Which of the following statements is true? a. If the Taylor series of the function f converges to f(x) for all x in some open interval containing a, then we say that the function f is analytic at x = a b. Every polynomial function is analytic everywhere. c. Every rational function is analytic wherever its denominator is nonzero. d. The sum of functions which are analytic at x = a is analytic at x = a. e. The product of functions which are analytic at x = a is analytic at x = a. f. All of the above. g. None of the above. Solution. Ans.: f. These are all true statements. (See page 196).

10 Page Analytic Functions Which of the following functions is analytic at x =? a. x+ x 4 x b. x 4 c. x+ x 4 d. x x 4 e. x+ x +4 f. None of the above Solution. Ans.: e. Choices (a)-(f) all have a singularity at x = because the denominator is zero there. I also accepted (f): None of the above as an answer. Several students pointed out that the function is not differentiable at 0, thus not analytic there.

11 Page Radius of Convergence Suppose we apply the method of power series to a DE and find out that the recurrence relation for the coefficients of n=0 is c n+1 = n+ c 3(n+1) n for n 0. What is the radius of convergence of the power series solution? a. 1 b. c. 3 d. 9 e. 0 f. None of the above Solution. Ans.: c. This question is based on Example, page 0. Note that c n ρ = lim n = 3. c n+1

12 Page Manipulating Power Series Consider the following power series. I. n=0 c nx n II. n=3 c nn(n 1)(n )x n 3 III. n=0 c nn(n 1)(n )x n 3 IV. n=1 c nx n Which of these power series represents the same function? a. I and IV only b. II and III only c. I and II only d. II and IV only e. All the above f. None of the above, or some other combination of I, II, III and IV Solution. Ans.: b.

13 Page Recurrence Relation Suppose c n+1 = n+1 c n for n 0. Which of the following is a closed form expression for c n? a. c n = n n! c o, n 0 b. c n = ( 1) n n (n+1)! c o, n 0 c. c n = ( 1) n n n! c 0, n 1 d. c n = ( 1) n n n! c 1, n 0 e. c n = ( 1) n n n! c o, n 0 f. None of the above. Solution. Ans.: c. 3.1 Example 1. I also accepted (f): None of the above. Several students pointed out that the formula is true for n 0, not just n 1.

14 Page Solving DEs with Power Series Solve: y =y. What is the radius of convergence? a. y = c 0 sin x; ρ = b. y = c 0 sin x; ρ = c. y = c 0 e x ; ρ = 1 d. y = c 0 e x ; ρ = e. No power series solution since ρ = 0 f. None of the above Solution. Ans.: d. This is 3.1 #1.

15 Page Solving IVPs with Power Series Solve the IVP: y + 4y = 0; y(0) = 0, y (0) = 3 a. sin x 3 sin x b. 3 c. 3 3 d. e. 3 cos x cos x sin 3x f. None of the above Solution. Ans.: b. 3.1 #19.

16 WRITTEN PROBLEM SHOW YOUR WORK Name: ID: Section: Note: You will be graded on the readability of your work. Use the back of this sheet, if necessary. 15. Power Series Solutions Solve the equation (x 1)y + y using the power series method. Solution. Ans. y(x) = c x This is 3.1 #7.

17 WRITTEN PROBLEM SHOW YOUR WORK Name: ID: Section: Note: You will be graded on the readability of your work. Use the back of this sheet, if necessary. 16. Power Series Solutions Show that the equation x y + x y + y = 0 has no nontrivial power series solution of the form y = n=0 c nx n. (Hint: Assume that the equation does have a power series and work it out. What happens?) Solution. Assume that there is a power series of the form y = n=0 c nx n. Then 0 = x y + x y + y = x c n n(n 1)x n + x = = n= c n n(n 1)x n + n= c n n(n 1)x n + n= = c 0 + c 1 x + = c 0 + c 1 x + n=1 c n nx n+1 + n=1 c n nx n 1 + c n x n n=0 c n x n n=0 c n nx n+1 + c 0 + c 1 x + n=1 c n x n n= [(n(n 1) + 1)c n + (n 1)c n 1 ]x n n= [(n n + 1)c n + (n 1)c n 1 ]x n. n= Therefore, c 0 = c 1 = 0 and c n = n 1 c n n+1 n 1 for n. Consequently, c n = 0 for all n 0, and so there is no nontrivial power series solution.