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# Math 285 Previous year bundle MATH 285

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This 330 page Bundle was uploaded by Tom Smith on Thursday September 8, 2016. The Bundle belongs to MATH 285 at University of Illinois at Urbana-Champaign taught by in Fall 2016. Since its upload, it has received 4 views. For similar materials see Intro to Differential Equations in Math at University of Illinois at Urbana-Champaign.

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NAME: NetID: MATH 285 E1/F1 Exam 1 (B) September 19, 2014 Instructor: Pascale▯ Problem Possible Actual 1 20 2 20 3 20 INSTRUCTIONS: 4 20 ▯ Do all work on these sheets. 5 20 ▯ Show all work. Total 100 1 1. (20 points) Consider the di▯erential equation dy = ▯xy dx Which of the following graphs could be a solution curve of this equation? Circle all that apply. 2 2. (20 points) An object moves along a one-dimensional axis. Its motion is descibed by a function x(t). It is subjected to an acceleration given by a(t) = 2 + 2▯ sin(▯t): Suppose that at t = 0, the velocity is zero: v(0) = 0. What is the net change in position between t = 0 and t = 1? That is, what is x(1) ▯ x(0)? 3 3. (20 points) Find the general solution, valid for x > 0, of dy 2y + x3 dx = x Hint: Linear equation, integrating factor. 4 4. (20 points) Consider the equation dy 2 dx ▯ 2y = xy Use the substitution u = y▯1 to transform this equation into a linear equation for u. Do not solve the resulting equation; the purpose of this problem is merely to transform the original equation for y into one for u. 5 5. (20 points) A metal ball has been heated to 2000 C. It is placed into a bath of ice water ▯ ▯10 ▯ at 0 C. After 10 seconds, it has cooled to a temperature of (2000e ) C (approximately 0:091 C). Suppose now that the metal ball is heated again to 2000 C, but instead it is placed into ▯ ▯ boiling water at 100 C. How long will it take to reach a temperature of 200 C? In both situation, the cooling process is governed by Newton’s law of cooling: dT = ▯k(T ▯ A) dt where A is the temperature of the water, and k is a constant. 6 This page is for work that doesn’t ▯t on the other pages. Please indicate the problem that the work goes with. 7 NAME: NetID: MATH 285 E1/F1 Exam 1 (A) September 19, 2014 Instructor: Pascale▯ Problem Possible Actual 1 20 2 20 3 20 INSTRUCTIONS: 4 20 ▯ Do all work on these sheets. 5 20 ▯ Show all work. Total 100 1 1. (20 points) Consider the di▯erential equation dy = xy dx Which of the following graphs could be a solution curve of this equation? Circle all that apply. 2 2. (20 points) An object moves along a one-dimensional axis. Its motion is descibed by a function x(t). It is subjected to an acceleration given by a(t) = 1 + ▯ sin(▯t): Suppose that at t = 0, the velocity is zero: v(0) = 0. What is the net change in position between t = 0 and t = 1? That is, what is x(1) ▯ x(0)? 3 3. (20 points) Find the general solution, valid for x > 0, of dy x + 2y dx = x Hint: Linear equation, integrating factor. 4 4. (20 points) Consider the equation dy 2 2 dx ▯ x y = y Use the substitution u = y to transform this equation into a linear equation for u. Do not solve the resulting equation; the purpose of this problem is merely to transform the original equation for y into one for u. 5 5. (20 points) A metal ball has been heated to 1000 C. It is placed into a bath of ice water ▯ ▯10 ▯ at 0 C. After 5 seconds, it has cooled to a temperature of (1000e ) C (approximately 0:045 C). Suppose now that the metal ball is heated again to 1000 C, but instead it is placed into ▯ ▯ boiling water at 100 C. How long will it take to reach a temperature of 200 C? In both situation, the cooling process is governed by Newton’s law of cooling: dT = ▯k(T ▯ A) dt where A is the temperature of the water, and k is a constant. 6 This page is for work that doesn’t ▯t on the other pages. Please indicate the problem that the work goes with. 7 NAME: NetID: MATH 285 E1/F1 Exam 2 (A) October 17, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Oct. 17, 2014. Total 100 1 1. (20 points) Let P(t) denote the population of a penguin colony in Antarctica. We assume that each female lays one egg each year, so the birth rate is 0:5 births per penguin per year. Due to scare resources, the death rate depends on the population as :05 + :001P deaths per penguin per year. Suppose that the penguin population is in equilibrium, meaning that it is constant in time: P(t) = P . What are the possible values of the equilibrium (constant) population P ? 0 0 2 x +5 3x x2 2. (20 points) Show that the functions f(x) = e and g(x) = e and h(x) = 4e are not linearly independent. That is, ▯nd constants A;B;C such that Af(x) + Bg(x) + Ch(x) = 0 for all x 3 3. (20 points) For each polynomial di▯erential operator p(D), ▯nd the solutions to the homo- d geneous di▯erential equation p(D)y = 0, where D = dx. It is not necessary to rederive the solution completely, but keep in mind that partial credit can be given if substantial work is shown. In the ▯rst three parts, you are asked to ▯nd the general real (not complex) solution. In the last part, you are asked to ▯nd one complex solution. 2 (a) p(D) = D + 3D + 2. Find the general real solution of p(D)y = 0. (b) p(D) = (D ▯ 4)(D + 3) . Find the general real solution of p(D)y = 0. 4 2 (c) p(D) = D ▯ D + 4. Find the general real solution of p(D)y = 0. p (d) p(D) = D ▯ 3i, where i = ▯1. In this case, there are no real solutions. Find one complex solution of p(D)y = 0. 5 4. (20 points) A mass is attached to a spring and a dashpot, so that its position x(t) obeys the di▯erential equation d x dx m 2 + c + kx = 0 dt dt The mass m, damping coe▯cient c, and spring constant k are given by m = 3;c = 6;k = 3. Suppose we do an experiment where the initial position and the initial velocity are x(0) = 1 and v(0) = ▯10. The function x(t) is determined by these initial conditions plus the di▯erential equation. How many times does the mass pass through the equilibrium position x = 0? That is, how many positive numbers t are there such that x(t) = 0? Note that if the solutions oscillate the answer could be in▯nitely many. 6 5. (20 points) Find the general solution of the di▯erential equation y ▯ 3y + 2y = 2x 7 NAME: NetID: MATH 285 E1/F1 Exam 2 (B) October 17, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Oct. 17, 2014. Total 100 1 1. (20 points) Let P(t) denote the population of a penguin colony in Antarctica. We assume that each female lays one egg each year, so the birth rate is 0:5 births per penguin per year. Due to scare resources, the death rate depends on the population as :01 + :0001P deaths per penguin per year. Suppose that the penguin population is in equilibrium, meaning that it is constant in time: P(t) = P . What are the possible values of the equilibrium (constant) population P ? 0 0 2 x2 5x 1+x2 2. (20 points) Show that the functions f(x) = 2e and g(x) = e and h(x) = e are not linearly independent. That is, ▯nd constants A;B;C such that Af(x) + Bg(x) + Ch(x) = 0 for all x 3 3. (20 points) For each polynomial di▯erential operator p(D), ▯nd the solutions to the homo- d geneous di▯erential equation p(D)y = 0, where D = dx. It is not necessary to rederive the solution completely, but keep in mind that partial credit can be given if substantial work is shown. In the ▯rst three parts, you are asked to ▯nd the general real (not complex) solution. In the last part, you are asked to ▯nd one complex solution. 2 (a) p(D) = D ▯ 4D + 3. Find the general real solution of p(D)y = 0. (b) p(D) = (D ▯ 3) (D + 2). Find the general real solution of p(D)y = 0. 4 2 (c) p(D) = D + 3D + 5. Find the general real solution of p(D)y = 0. p (d) p(D) = D ▯ 5i, where i = ▯1. In this case, there are no real solutions. Find one complex solution of p(D)y = 0. 5 4. (20 points) A mass is attached to a spring and a dashpot, so that its position x(t) obeys the di▯erential equation d x dx m 2 + c + kx = 0 dt dt The mass m, damping coe▯cient c, and spring constant k are given by m = 4;c = 8;k = 4. Suppose we do an experiment where the initial position and the initial velocity are x(0) = ▯1 and v(0) = 10. The function x(t) is determined by these initial conditions plus the di▯erential equation. How many times does the mass pass through the equilibrium position x = 0? That is, how many positive numbers t are there such that x(t) = 0? Note that if the solutions oscillate the answer could be in▯nitely many. 6 5. (20 points) Find the general solution of the di▯erential equation y + 5y + 6y = 3x 7 NAME: NetID: MATH 285 E1/F1 Exam 2 (C) October 17, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Oct. 17, 2014. Total 100 1 1. (20 points) Let P(t) denote the population of a penguin colony in Antarctica. We assume that each female lays one egg each year, so the birth rate is 0:5 births per penguin per year. Due to scare resources, the death rate depends on the population as :03 + :01P deaths per penguin per year. Suppose that the penguin population is in equilibrium, meaning that it is constant in time: P(t) = P . What are the possible values of the equilibrium (constant) population P ? 0 0 2 5x x +1 x2 2. (20 points) Show that the functions f(x) = e and g(x) = e and h(x) = 4e are not linearly independent. That is, ▯nd constants A;B;C such that Af(x) + Bg(x) + Ch(x) = 0 for all x 3 3. (20 points) For each polynomial di▯erential operator p(D), ▯nd the solutions to the homo- d geneous di▯erential equation p(D)y = 0, where D = dx. It is not necessary to rederive the solution completely, but keep in mind that partial credit can be given if substantial work is shown. In the ▯rst three parts, you are asked to ▯nd the general real (not complex) solution. In the last part, you are asked to ▯nd one complex solution. 2 (a) p(D) = D + 4D + 3. Find the general real solution of p(D)y = 0. (b) p(D) = (D ▯ 2)(D + 6) . Find the general real solution of p(D)y = 0. 4 2 (c) p(D) = D ▯ D + 5. Find the general real solution of p(D)y = 0. p (d) p(D) = D ▯ 2i, where i = ▯1. In this case, there are no real solutions. Find one complex solution of p(D)y = 0. 5 4. (20 points) A mass is attached to a spring and a dashpot, so that its position x(t) obeys the di▯erential equation d x dx m 2 + c + kx = 0 dt dt The mass m, damping coe▯cient c, and spring constant k are given by m = 2;c = 4;k = 2. Suppose we do an experiment where the initial position and the initial velocity are x(0) = 1 and v(0) = ▯10. The function x(t) is determined by these initial conditions plus the di▯erential equation. How many times does the mass pass through the equilibrium position x = 0? That is, how many positive numbers t are there such that x(t) = 0? Note that if the solutions oscillate the answer could be in▯nitely many. 6 5. (20 points) Find the general solution of the di▯erential equation y ▯ 6y + 8y = 4x 7 NAME: NetID: MATH 285 E1/F1 Exam 3 (A) November 14, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Nov. 14, 2014. Total 100 1 Orthogonality formulas Z L ( m▯t n▯t 0; m 6 n cos L cos L dt = (1) ▯L L; m = n ( Z L m▯t n▯t 0; m 6 n sin sin dt = (2) ▯L L L L; m = n Z L m▯t n▯t cos sin dt = 0 (3) ▯L L L Some integral formulas Z ucosudu = usinu + cosu + C (4) Z usinudu = ▯ucosu + sinu + C (5) 2 1. (20 points) Find the general solution of the di▯erential equation y ▯ 3y = xe 3x 3 2. (20 points) Consider the forced oscillator with mass m = 1, spring constant k = 10, no damping c = 0, and forcing function F(t): F(t) = sin2t + 2sin4t + cos6t Find a particular solution of the di▯erential equation mx + kx = F(t). 4 3. (a) (10 points) Suppose that a function f(t) which is periodic of period 2▯ has the Fourier series 1 X (▯1)n+1 f(t) = sinnt n=1 n(n + 1) Use the orthogonality formulas to evaluate the integral Z ▯ f(t)sin4tdt ▯▯ (b) (10 points) Let g(t) be the function which is periodic of period 30, and which is de▯ned on the interval ▯15 ▯ t < 15 by the formula g(t) = 2 + 3t + 6t Set up, but do not evaluate, an integral expression for the coe▯cient of coin the 15 Fourier series of g(t) (also known 3s a in our standard notation). 5 4. (a) (5 points) Consider the function which is periodic of period 2▯ de▯ned on the interval ▯▯ ▯ t < ▯ 8 >16; ▯▯ ▯ t < 0 < f(t) = 609250; t = 0 : t; 0 < t < ▯ If we take the Fourier series of f(t), and put t = 0 in that series, what number does it converge to? Put another way, what is the sum of the Fourier series of f(t) at t = 0? Explain your answer (brie y). (b) (15 points) Consider the function de▯ned by the Fourier series X1 g(t) = 3e▯2nsinn▯t n=1 R Find a Fourier series expression for the antig(t)dt. You are not expected to address the question of convergence. 6 5. (20 points) Find the Fourier series of the periodic function of period 2 de▯ned on the interval ▯1 ▯ t < 1 by f(t) = 2jtj; ▯1 ▯ t < 1 Hint: You should use the fact that f(t) is an even function. 7 This page is for work that doesn’t ▯t on other pages. 8 NAME: NetID: MATH 285 E1/F1 Exam 3 (A) November 14, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Nov. 14, 2014. Total 100 1 Orthogonality formulas Z L ( m▯t n▯t 0; m 6 n cos L cos L dt = (1) ▯L L; m = n ( Z L m▯t n▯t 0; m 6 n sin sin dt = (2) ▯L L L L; m = n Z L m▯t n▯t cos sin dt = 0 (3) ▯L L L Some integral formulas Z ucosudu = usinu + cosu + C (4) Z usinudu = ▯ucosu + sinu + C (5) 2 1. (20 points) Find the general solution of the di▯erential equation y ▯ 3y = xe 3x 3 2. (20 points) Consider the forced oscillator with mass m = 1, spring constant k = 10, no damping c = 0, and forcing function F(t): F(t) = sin2t + 2sin4t + cos6t Find a particular solution of the di▯erential equation mx + kx = F(t). 4 3. (a) (10 points) Suppose that a function f(t) which is periodic of period 2▯ has the Fourier series 1 X (▯1)n+1 f(t) = sinnt n=1 n(n + 1) Use the orthogonality formulas to evaluate the integral Z ▯ f(t)sin4tdt ▯▯ (b) (10 points) Let g(t) be the function which is periodic of period 30, and which is de▯ned on the interval ▯15 ▯ t < 15 by the formula g(t) = 2 + 3t + 6t Set up, but do not evaluate, an integral expression for the coe▯cient of coin the 15 Fourier series of g(t) (also known 3s a in our standard notation). 5 4. (a) (5 points) Consider the function which is periodic of period 2▯ de▯ned on the interval ▯▯ ▯ t < ▯ 8 >16; ▯▯ ▯ t < 0 < f(t) = 609250; t = 0 : t; 0 < t < ▯ If we take the Fourier series of f(t), and put t = 0 in that series, what number does it converge to? Put another way, what is the sum of the Fourier series of f(t) at t = 0? Explain your answer (brie y). (b) (15 points) Consider the function de▯ned by the Fourier series X1 g(t) = 3e▯2nsinn▯t n=1 R Find a Fourier series expression for the antig(t)dt. You are not expected to address the question of convergence. 6 5. (20 points) Find the Fourier series of the periodic function of period 2 de▯ned on the interval ▯1 ▯ t < 1 by f(t) = 2jtj; ▯1 ▯ t < 1 Hint: You should use the fact that f(t) is an even function. 7 This page is for work that doesn’t ▯t on other pages. 8 NAME: NetID: MATH 285 E1/F1 Exam 3 (B) November 14, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Nov. 14, 2014. Total 100 1 Orthogonality formulas Z L ( m▯t n▯t 0; m 6 n cos L cos L dt = (1) ▯L L; m = n ( Z L m▯t n▯t 0; m 6 n sin sin dt = (2) ▯L L L L; m = n Z L m▯t n▯t cos sin dt = 0 (3) ▯L L L Some integral formulas Z ucosudu = usinu + cosu + C (4) Z usinudu = ▯ucosu + sinu + C (5) 2 1. (20 points) Find the general solution of the di▯erential equation y ▯ 5y = xe 5x 3 2. (20 points) Consider the forced oscillator with mass m = 1, spring constant k = 5, no damping c = 0, and forcing function F(t): F(t) = cos2t + sin4t + 3sin6t Find a particular solution of the di▯erential equation mx + kx = F(t). 4 3. (a) (10 points) Suppose that a function f(t) which is periodic of period 2▯ has the Fourier series 1 X (▯1)n f(t) = 2 sinnt n=1 n + 1 Use the orthogonality formulas to evaluate the integral Z ▯ f(t)sin3tdt ▯▯ (b) (10 points) Let g(t) be the function which is periodic of period 20, and which is de▯ned on the interval ▯10 ▯ t < 10 by the formula g(t) = 7t + 2t + 3 Set up, but do not evaluate, an integral expression for the coe▯cient of siin the 10 Fourier series of g(t) (also known 3s b in our standard notation). 5 4. (a) (5 points) Consider the function which is periodic of period 2▯ de▯ned on the interval ▯▯ ▯ t < ▯ 8 >t; ▯▯ ▯ t < 0 < f(t) = 609250; t = 0 : 8; 0 < t < ▯ If we take the Fourier series of f(t), and put t = 0 in that series, what number does it converge to? Put another way, what is the sum of the Fourier series of f(t) at t = 0? Explain your answer (brie y). (b) (15 points) Consider the function de▯ned by the Fourier series X1 g(t) = 4e▯4nsinn▯t n=1 R Find a Fourier series expression for the antig(t)dt. You are not expected to address the question of convergence. 6 5. (20 points) Find the Fourier series of the periodic function of period 2 de▯ned on the interval ▯1 ▯ t < 1 by f(t) = 4jtj; ▯1 ▯ t < 1 Hint: You should use the fact that f(t) is an even function. 7 This page is for work that doesn’t ▯t on other pages. 8 NAME: NetID: MATH 285 E1/F1 Exam 3 (B) November 14, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Nov. 14, 2014. Total 100 1 Orthogonality formulas Z L ( m▯t n▯t 0; m 6 n cos L cos L dt = (1) ▯L L; m = n ( Z L m▯t n▯t 0; m 6 n sin sin dt = (2) ▯L L L L; m = n Z L m▯t n▯t cos sin dt = 0 (3) ▯L L L Some integral formulas Z ucosudu = usinu + cosu + C (4) Z usinudu = ▯ucosu + sinu + C (5) 2 1. (20 points) Find the general solution of the di▯erential equation y ▯ 5y = xe 5x 3 2. (20 points) Consider the forced oscillator with mass m = 1, spring constant k = 5, no damping c = 0, and forcing function F(t): F(t) = cos2t + sin4t + 3sin6t Find a particular solution of the di▯erential equation mx + kx = F(t). 4 3. (a) (10 points) Suppose that a function f(t) which is periodic of period 2▯ has the Fourier series 1 X (▯1)n f(t) = 2 sinnt n=1 n + 1 Use the orthogonality formulas to evaluate the integral Z ▯ f(t)sin3tdt ▯▯ (b) (10 points) Let g(t) be the function which is periodic of period 20, and which is de▯ned on the interval ▯10 ▯ t < 10 by the formula g(t) = 7t + 2t + 3 Set up, but do not evaluate, an integral expression for the coe▯cient of siin the 10 Fourier series of g(t) (also known 3s b in our standard notation). 5 4. (a) (5 points) Consider the function which is periodic of period 2▯ de▯ned on the interval ▯▯ ▯ t < ▯ 8 >t; ▯▯ ▯ t < 0 < f(t) = 609250; t = 0 : 8; 0 < t < ▯ If we take the Fourier series of f(t), and put t = 0 in that series, what number does it converge to? Put another way, what is the sum of the Fourier series of f(t) at t = 0? Explain your answer (brie y). (b) (15 points) Consider the function de▯ned by the Fourier series X1 g(t) = 4e▯4nsinn▯t n=1 R Find a Fourier series expression for the antig(t)dt. You are not expected to address the question of convergence. 6 5. (20 points) Find the Fourier series of the periodic function of period 2 de▯ned on the interval ▯1 ▯ t < 1 by f(t) = 4jtj; ▯1 ▯ t < 1 Hint: You should use the fact that f(t) is an even function. 7 This page is for work that doesn’t ▯t on other pages. 8 NAME: NetID: MATH 285 E1/F1 Exam 3 (C) November 14, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Nov. 14, 2014. Total 100 1 Orthogonality formulas Z L ( m▯t n▯t 0; m 6 n cos L cos L dt = (1) ▯L L; m = n ( Z L m▯t n▯t 0; m 6 n sin sin dt = (2) ▯L L L L; m = n Z L m▯t n▯t cos sin dt = 0 (3) ▯L L L Some integral formulas Z ucosudu = usinu + cosu + C (4) Z usinudu = ▯ucosu + sinu + C (5) 2 1. (20 points) Find the general solution of the di▯erential equation y ▯ 4y = xe 4x 3 2. (20 points) Consider the forced oscillator with mass m = 1, spring constant k = 7, no damping c = 0, and forcing function F(t): F(t) = 2sin2t + cos4t + cos6t Find a particular solution of the di▯erential equation mx + kx = F(t). 4 3. (a) (10 points) Suppose that a function f(t) which is periodic of period 2▯ has the Fourier series 1 X (▯1)n+1 f(t) = 2 sinnt n=1 n + 3 Use the orthogonality formulas to evaluate the integral Z ▯ f(t)sin2tdt ▯▯ (b) (10 points) Let g(t) be the function which is periodic of period 24, and which is de▯ned on the interval ▯12 ▯ t < 12 by the formula g(t) = 4 + 2t + 7t Set up, but do not evaluate, an integral expression for the coe▯cient of coin the 12 Fourier series of g(t) (also known 5s a in our standard notation). 5 4. (a) (5 points) Consider the function which is periodic of period 2▯ de▯ned on the interval ▯▯ ▯ t < ▯ 8 >800; ▯▯ ▯ t < 0 < f(t) = 609250; t = 0 : t; 0 < t < ▯ If we take the Fourier series of f(t), and put t = 0 in that series, what number does it converge to? Put another way, what is the sum of the Fourier series of f(t) at t = 0? Explain your answer (brie y). (b) (15 points) Consider the function de▯ned by the Fourier series X1 g(t) = 2e▯3nsinn▯t n=1 R Find a Fourier series expression for the antig(t)dt. You are not expected to address the question of convergence. 6 5. (20 points) Find the Fourier series of the periodic function of period 2 de▯ned on the interval ▯1 ▯ t < 1 by f(t) = 3jtj; ▯1 ▯ t < 1 Hint: You should use the fact that f(t) is an even function. 7 This page is for work that doesn’t ▯t on other pages. 8 NAME: NetID: MATH 285 E1/F1 Exam 3 (C) November 14, 2014 Instructor: Pascale▯ Problem Possible Actual INSTRUCTIONS: 1 20 ▯ Do all work on these sheets. 2 20 ▯ Show all work. 3 20 ▯ The exam is 50 minutes. 4 20 ▯ Do not discuss this exam with anyone 5 20 until after 3:00 pm on Nov. 14, 2014. Total 100 1 Orthogonality formulas Z L ( m▯t n▯t 0; m 6 n cos L cos L dt = (1) ▯L L; m = n ( Z L m▯t n▯t 0; m 6 n sin sin dt = (2) ▯L L L L; m = n Z L m▯t n▯t cos sin dt = 0 (3) ▯L L L Some integral formulas Z ucosudu = usinu + cosu + C (4) Z usinudu = ▯ucosu + sinu + C (5) 2 1. (20 points) Find the general solution of the di▯erential equation y ▯ 4y = xe 4x 3 2. (20 points) Consider the forced oscillator with mass m = 1, spring constant k = 7, no damping c = 0, and forcing function F(t): F(t) = 2sin2t + cos4t + cos6t Find a particular solution of the di▯erential equation mx + kx = F(t). 4 3. (a) (10 points) Suppose that a function f(t) which is periodic of period 2▯ has the Fourier series 1 X (▯1)n+1 f(t) = 2 sinnt n=1 n + 3 Use the orthogonality formulas to evaluate the integral Z ▯ f(t)sin2tdt ▯▯ (b) (10 points) Let g(t) be the function which is periodic of period 24, and which is de▯ned on the interval ▯12 ▯ t < 12 by the formula g(t) = 4 + 2t + 7t Set up, but do not evaluate, an integral expression for the coe▯cient of coin the 12 Fourier series of g(t) (also known 5s a in our standard notation). 5 4. (a) (5 points) Consider the function which is periodic of period 2▯ de▯ned on the interval ▯▯ ▯ t < ▯ 8 >800; ▯▯ ▯ t < 0 < f(t) = 609250; t = 0 : t; 0 < t < ▯ If we take the Fourier series of f(t), and put t = 0 in that series, what number does it converge to? Put another way, what is the sum of the Fourier series of f(t) at t = 0? Explain your answer (brie y). (b) (15 points) Consider the function de▯ned by the Fourier series X1 g(t) = 2e▯3nsinn▯t n=1 R Find a Fourier series expression for the antig(t)dt. You are not expected to address the question of convergence. 6 5. (20 points) Find the Fourier series of the periodic function of period 2 de▯ned on the interval ▯1 ▯ t < 1 by f(t) = 3jtj; ▯1 ▯ t < 1 Hint: You should use the fact that f(t) is an even function. 7 This page is for work that doesn’t ▯t on other pages. 8 MATH 285 E1/F1 GRADED HOMEWORK SET 1 DUE WEDNESDAY SEPTEMBER 10 IN LECTURE IT WOULD BE SO SWEET if you followed these instructions: Please put each problem on a separate sheet of paper with your name and section (E1 or F1). If a problem runs multiple pages, please staple all the pages for a single problem together. Think of each problem as a separate assignment. This may be annoying, but it will greatly streamline the grading process, resulting in faster feedback for you. Thank you! Section and problem numbers refer to Di▯erential Equations & Boundary Value Problems, Fourth Edition, by Edwards and Penney. (1) Let f(x) be the function de▯ned piece-wise as ( x if x ▯ 5 f(x) = 5 if x > 5 Find the solution of the initial value problem dy = f(x); y(0) = 100: dx Hint: Your solution will also be de▯ned piece-wise. (2) Consider the di▯erential equation dy x dx = ▯ y Sketch the slope ▯eld for this equation.What are the solution curves? Hint: You should recognize them as semi-familiar geometric shapes. (3) Section 1.4, problem 22. (4) Section 1.5, problem 10 (Find the general solution valid for x > 0). (5) Section 1.6, problem 14. 1 MATH 285 E1/F1 GRADED HOMEWORK SET 1 DUE WEDNESDAY SEPTEMBER 10 IN LECTURE IT WOULD BE SO SWEET if you followed these instructions: Please put each problem on a separate sheet of paper with your name and section (E1 or F1). If a problem runs multiple pages, please staple all the pages for a single problem together. Think of each problem as a separate assignment. This may be annoying, but it will greatly streamline the grading process, resulting in faster feedback for you. Thank you! Section and problem numbers refer to Di▯erential Equations & Boundary Value Problems, Fourth Edition, by Edwards and Penney. (1) Let f(x) be the function de▯ned piece-wise as ( x if x ▯ 5 f(x) = 5 if x > 5 Find the solution of the initial value problem dy dx = f(x); y(0) = 100: Hint: Your solution will also be de▯ned piece-wise. Solution: ( 1 2 y(x) = 100 + 2 if x ▯ 5 100 ▯ 25+ 5x if x > 5 2 Tdyobtain this, ▯rst consider the range x ▯ 5. The equation becomes dx= x, with general solution y(x) = 2 +C. In order to satisfy the initial condition y(0) = 100, the constant must be C = 100. This gives the ▯rst part of the piece-wise de▯nition. Second, consider the range x > 5. The equation becomes dy= 5, dx with general solution y(x) = 5x + D. We have to choose the value of D so that the two pieces match at x = 5 (that is, so that y(x) is a 1 2 continuous function). At x = 5, the formula 100+ x2has the value 100+ 25. So we need 5(5)+D = 100+ 25, hence D = 100▯ 25. This 2 2 2 gives the second part of the piece-wise de▯nition. (2) Consider the di▯erential equation dy x = ▯ : dx y Sketch the slope ▯eld for this equation. What are the solution curves? Hint: You should recognize them as semi-familiar geometric shapes. 1 MATH 285 E1/F1 GRADED HOMEWORK SET 1 DUE WEDNESDAY SEPTEMBER 10 IN LECTURE Solution: The slope ▯eld is perpendicular to the lines through the origin. The solution curves are the upper and lower half-circles centered at the origin. Note that an entire circle is not a solution curve because it does not de▯ne y as a function of x. (3) Section 1.4, problem 22. Solution: x ▯x y(x) = ▯3e : dy Starting from dx = 4x y ▯ y, separate variables to obtain Z Z dy = (4x ▯ 1)dx y 4 lnjyj = x ▯ x + C C x ▯4 x ▯x y = ▯e e = De C (where the constant D absorbs the plus/minus sign and e ). We now use the initial condition y(1) = ▯3. 1 ▯1 0 ▯3 = y(1) = De = De = D: So D = ▯3. (4) Section 1.5, problem 10 (Find the general solution valid for x > 0). Solution: y(x) = 3x + Cx 3=: We ▯rst put the equation 2xy ▯ 3y = 9x into standard form y ▯ 3 y = 9 x2 2x 2 An integrating factor is R ▯ 3 dx ▯ 3ln jxj ln jxj ▯3=2 ▯3=2 e 2x = e 2 = (e ) = jxj Since we are restricting the domain to x > 0, we may simply use ▯3=2 x as the integrating factor. Multiply through: ▯3=2 0 3 ▯5=2 9 1=2 x y ▯ 2 x y = 2x ; d ▯3=2 9 1=2 (x y) = x : dx 2 Integrate: x▯3=2y = 3x 3=2+ C: Solve for y: y = 3x + Cx 3=2: MATH 285 E1/F1 GRADED HOMEWORK SET 1DUE WEDNESDAY SEPTEMBER 10 IN LECTURE 3 (5) Section 1.6, problem 14. Solution: p y(x) = ▯ 2Cx + C : p Starting from yy +x = x + y , use the substitution u = x +y . Then u = 2x + 2yy , which we recognize as 2 times the left-hand side. The equation becomes 0 p (1=2)u = u: Separate variables: Z Z (1=2)u▯1=2du = dx; u1=2 = x + C; 2 u = (x + C) : Finally, solve for y: 2 2 2 x + y = u = (x + C) ; q y = ▯ (x + C) ▯ x ; p 2 y = ▯ 2Cx + C : MATH 285 E1/F1 GRADED HOMEWORK SET 2 DUE FRIDAY SEPTEMBER 26 IN LECTURE LET’S CHANGE IT UP WITH SOME FRESH NEW INSTRUC- TIONS: This time, the homework has two parts, A and B. Please turn in each part separately, with your name and section clearly marked on each part. Please staple all the pages for a particular part together, but do not staple the two parts to each other. When you turn the homework in, there will be two boxes. Thank you! Part A (1) (5 points) Consider the equation xy dy xy (2y + xe ) + x + ye = 0 dx Show that this equation is exact, and ▯nd an implicit equation for the solution. (2) (5 points) Let P(t) denote the number of technology startups in the San Francisco Bay area, where time t is measured in months. Startups are created when two recent graduates decide to found one, and they either fail or they are bought by Google. The rate at which startups are formed is given by (10▯0:01P)P per month. Also, each month, 10% of the startups fail, and 5% are bought by Google. How many startups do you expect to exist at a particular time many months into the future? That is, what is lim t!1 P(t)? Part B (3) (5 points) Suppose that y (x) and y (x) are two solutions of the 1 2 nonhomogeneous equation A(x)y + B(x)y + C(x)y = F(x): Prove that their di▯erence y (x) ▯ y (x) is a solution of the homo- 1 2 geneous equation 00 0 A(x)y + B(x)y + C(x)y = 0: (4) (15 points) (a) Find the general solution of the second order linear homoge- neous equation 4y + 8y + 3y = 0: 1 2 GRADED HOMEWORK 2 (b) Using your innate cleverness, ▯nd a particular solution to the nonhomongeneous equation 4y + 8y + 3y = 15: (c) Using the results of the previous two parts, ▯nd solution of the initial value problem 00 0 0 4y + 8y + 3y = 15; y(0) = 0; y (0) = 0: MATH 285 E1/F1 GRADED HOMEWORK SET 2 DUE FRIDAY SEPTEMBER 26 IN LECTURE LET’S CHANGE IT UP WITH SOME FRESH NEW INSTRUC- TIONS: This time, the homework has two parts, A and B. Please turn in each part separately, with your name and section clearly marked on each part. Please staple all the pages for a particular part together, but do not staple the two parts to each other. When you turn the homework in, there will be two boxes. Thank you! Part A (1) (5 points) Consider the equation xy dy xy (2y + xe ) dx + x + ye = 0 Show that this equation is exact, and ▯nd an implicit equation for the solution. dy The equation has the form M(x;y)+N(x;y) dx = 0 where M(x;y) = x+ye xyand N(x;y) = 2y+xe . We compare the partial derivatives @M xy xy = 0 + e + xye ; @y @N = 0 + exy+ xye xy @x Since these are the same, the equation is exact. Now we look for a function F(x;y) such that @F = x + ye ; @F = 2y + xe : @x @y Integrating the equation for the x-derivative gives Z xy 2 xy F(x;y) = (x + ye )dx = x =2 + e + C(y); where C(y) is some function of y only. Plugging this into the equa- tion for the y-derivative gives @ ▯ ▯ 2y + xexy = x =2 + exy + C(y) = xe xy+ C (y): @y Thus C (y) = 2y, and we can take C(y) = y . Thus F(x;y) = x =2 + e xy+ y works. The implicit equation for the solutions is then x =2 + exy+ y = C; where C is an arbitrary constant. 1 2 GRADED HOMEWORK 2 (2) (5 points) Let P(t) denote the number of technology startups in the San Francisco Bay area, where time t is measured in months. Startups are created when two recent graduates decide to found one, and they either fail or they are bought by Google. The rate at which startups are formed is given by (10▯0:01P)P per month. Also, each month, 10% of the startups fail, and 5% are bought by Google. How many startups do you expect to exist at a particular time many months into the future? That is, what is lim t!1 P(t)? The di▯erential equation for P(t) is dP dt = (10 ▯ 0:01P)P ▯ 0:1P ▯ 0:05P; where the ▯rst term is the number of startups formed per month, the term 0:1P is the number of startups that fail per month, and 0:05P is the number of startups that are bought by Google per month. We can write this as a logistic equation: dP = (9:85 ▯ 0:01P)P = 0:01P(985 ▯ P) dt dP We know that for the logistic equation dt = kP(M ▯ P), the value of P converges to M in the long run. So in this case, the number of startups will converge to 985 in the long run, that is, lim t!1 P(t) = 985. Part B (3) (5 points) Suppose that y (x) an1 y (x) are 2wo solutions of the nonhomogeneous equation 00 0 A(x)y + B(x)y + C(x)y = F(x): Prove that their di▯erence y (x) 1 y (x) i2 a solution of the homo- geneous equation 00 0 A(x)y + B(x)y + C(x)y = 0: Let us abbreviate A(x) as just A, and so on. We need to consider the expression 00 0 Z = A(y ▯ 1 ) + 2(y ▯ y ) +1C(y 2 y ) 1 2 and show that it is zero. First expand out the terms, using the facts 0 0 0 00 00 00 (y1▯ y )2= y ▯ 1 ; 2 (y1▯ y )2= y ▯ y1 2 to get Z = Ay ▯1Ay + By2▯ By + 1y ▯ Cy 2 1 2 Then collect the terms involving y and y to get

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