Work sheet / Things to know. Chapter 3

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1 MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients only. What is the general form of a second order linear equation with constant coefficients? Ex Can you think of any function(s) that satisfy each equation (w/ constant coefficients) below? (a) y 25y = 0 (b) y + 25y = 0 (c) y 25 y = 0 The example (c) above is an instance of a second order linear equation with the y-term missing. It is essentially a first order linear equation in disguise. All equations of this type can be solved by changing it into a first order equation with the substitutions u = y and u = y, then use the integrating factor method to solve for u, and integrate the result to find y. 1

2 2. The characteristic equation Given the equation ay + by + cy = 0, a 0, what is its characteristic equation? Any root, r, of the characteristic equation has the property that y = e rt always satisfies the equation above. Therefore, y = e rt will be a particular solution for each root r. Consequently, an important formula to remember for this class is (surprisingly) the quadratic formula: r = b± b 2 4ac 2a Note that the characteristic equation method does not require the given differential equation to be put into its standard form first the quadratic formula simply doesn t care whether or not the leading coefficient is 1. Suppose r 1 and r 2 are two distinct real roots of the characteristic equation, what is the general solution of the differential equation? y = Ex y + y 12y = 0 What is its characteristic equation? What are the roots of the characteristic equation? Based on the roots, what are 2 particular solutions of the equation? The general solution is y = Ex y + 3y 2y = 0 y = 2

3 3. Initial Value Problems What do the initial conditions of a second order differential equation look like? A second order equation s general solution will have 2 arbitrary constants / coefficients. Therefore, an IVP will have 2 initial conditions in order to give 2 (algebraic) equations needed to solve for them. What must the conditions look like? Ex Take the previous example y + y 12y = 0. Find its solution satisfying (a) y(0) = 0, y (0) = 2 (b) y(0) = 2, y (0) = 6 How to (easily) work with initial conditions where t 0 0? Ex y + y 12y = 0, y(45000) = 0, y (45000) = 2 3

4 The Existence and Uniqueness Theorem (for second order linear equations) It is really the same theorem as the one we saw earlier, except this one is in the context of second order linear equation. What does it say? How to find the largest interval (that is, the interval of validity) on which a particular solution is guaranteed to exist uniquely? Ex Consider the equation given below. For each set of initial conditions, find the largest interval on which the particular solution is guaranteed to exist uniquely. (t 2 + 2t 8) y + sin(2t) y + (t + 9) 3 y = t 1 e 5t (a) y(3) = 1, (b) y( 1) = 9π, y (3) = 9 y ( 1) = e 1 4

5 4. The general solution of second order linear equations Structure of a general solution: homogeneous second order linear equation: y = A nonhomogeneous equation, however, will have a slightly different form of solution. Wronskian know the determinant formula W(y 1, y 2 )(t) = *Note that the Wronskian is defined as a function of t. Fundamental solutions What are they? Why are they important? How to determine if any two given solutions are fundamental? Ex Which pair(s) of functions below can be fundamental solutions? (1) 10, t + 2 (2) e 2t, e 2t (3) e t, 0 (4) 2e 3t, e 3t + 2 (5) cos 6t, 2sin 6t (6) 5sin t, cos(t + π/2) (7) cos 2t, sin(2t 2π) Ex Given that y 1 = te t and y 2 = 2te t 7ln(t) are both known solutions of a certain equation y + p(t)y + q(t)y = 0. What is the general solution of this equation? 5

6 5. The Abel s Theorem It gives us a way to determine, up to a constant multiplier, the Wronskian of any pair of solutions of a given second order homogeneous linear equation, by looking at just the equation itself only. What does it say: W(y 1, y 2 )(t) = (Due to their similarity) do not confuse this formula with another formula we have seen before. Which formula? Ex (Final Exam, fall 2008) Let y 1 (t) and y 2 (t) be any two solutions of the second order linear equation t 2 y 6t y + cos(3t)y = 0. What is the general form of their Wronskian, W(y 1, y 2 )(t)? Ex (Exam I, spring 2007) Consider the second order linear differential equation t y 2y + y = 0. Suppose y 1 (t) and y 2 (t) are two fundamental solutions of the equation such that y 1 (1) = 2, y 1 (1) = 0, y 2 (1) = 2, and y 2 (1) = 2. Compute their Wronskian W(y 1, y 2 )(t). 6

7 6. Characteristic equation: complex roots case (Background info) Euler s formula: Suppose r = λ ± µi are two complex conjugate roots of the characteristic equation, what is the general solution of the differential equation? y = Ex y 4y + 40y = 0 What are the roots of the characteristic equation? The general solution is y = What is the particular solution satisfying y(0) = 1 and y (0) = 6? What is the particular solution satisfying y(755) = 1 and y (755) = 6? 7

8 7. Characteristic equation: repeated real root case Suppose r is a repeated real root of the characteristic equation, what is the general solution of the differential equation? y = Ex y + 12y + 36y = 0 What are the roots of the characteristic equation? The general solution is y = What is the particular solution satisfying y(0) = 3 and y (0) = 3? What is the particular solution satisfying y(6789) = 3 and y (6789) = 3? 8

9 8. Reduction of order The characteristic equation method only works for equations of constant coefficients. While we have not (and will not, in this course) learned a technique to solve a generic second order linear equation with non-constant coefficients, we nevertheless have learned enough to be able to solve such an equation of non-constant coefficients provided that we know one nonzero solution of the equation already. This type of problems is called reduction of order. Why is it called reduction of order? There are more than one ways to solve such a problem. What we learn in class is a synthetic method. It is different, and easier, than the method in the textbook. What are the key steps of our method? Ex Given that y 1 = t is a solution, find the general solution of the equation t 2 y + 2t y 2y = 0. Find its general solution. 9

10 9. Nonhomogeneous second order linear differential equation Standard form (constant coefficients): What is its corresponding homogeneous equation? Structure of its general solution: y = What is the complementary / homogeneous solution? What is the particular / nonhomogeneous solution? Note that the different context under which the name particular solution is used, as compared to the solution of an IVP. 10. Method of Undetermined Coefficients What is the idea behind the method? You should ALWAYS solve for the complementary solution first. The starting choices: (1) If g(t) is an exponential function. Ex Given g(t) = 3e 5t, choose Y = 10

11 (2) If g(t) is a polynomial, or a power of t. Ex Given g(t) = 2t t 3 + t 4, choose Y = (3) If g(t) is a sinusoidal function, sine or cosine.. Ex Given g(t) = 9sin 3πt, choose Y = If g(t) is a sum or difference of the basic function, how do you choose Y? Ex (a) Given g(t) = 3e 5t 9sin 3πt, choose Y = (b) Given g(t) = t 2 + 2cos 4t, choose Y = The possible glitch with our starting choices, how to spot it, and how to correct it When do you need to multiply your starting choice by t (or by t 2 )? AKA, why you should always solve for the complementary solution first. How to spot the problem? Ex Consider the equation y + y 12y = g(t) What is its complementary solution? (a) Given g(t) = 9e 4t, choose Y = (b) Given g(t) = 2e 3t 4e 3t, choose Y = 11

12 Ex Use the method of undetermined coefficients to solve y + y 12y = 9e 4t. How to choose Y(t) when g(t) is a product of elementary functions? What to do when g(t) is a product of any two, or all three, of polynomial/power, exponential, and sinusoidal functions? The principles: (i) If g(t) is a sum of several products, do each part separately. (Review the rule about handling g(t) being a sum/difference.) (ii) The starting choice for Y shall be a product of the corresponding starting choice for each component function in g(t). Every possible term in the resulting product of the basic choice of functions must be represented. (iii) Each of the terms in the starting choice shall have its own unique coefficient. (iv) Lastly, the starting choice must be checked against y c for identical terms. If any shared term is found, then every term in the starting choice must be multiplied by t. Repeat until no shared term is found. Know well how to apply the above principles. Do many exercises! 12

13 Ex (Exam 1, fall 2007) Determine the most suitable choice of Y(t) for each equation. (a) y 4y + 8y = 2e 2t 5t 2 + sin 2t (b) y 4y + 8y = e 2t sin 2t + 1 (c) y 4y + 8y = t 2 e t cos 5t Ex (Exam 2, fall 2001) Solve the initial value problem y + 4y = e t, y(0) = 0, y (0) = 0. 13

14 Ex Use the method of undetermined coefficients to solve y + 12y + 36y = t + 3 2e 6t. Here are a couple exercises to test your familiarity with some of the concepts how the different parts are put together to give the solution of a second order linear equation of this chapter thus far. Review Ex 3.1 Given that y = 3e 5t is a known solution of the equation y + 12y + 36y = g(t). What is the general solution of this equation? Review Ex 3.2 Given that y 1 = 2πe 5t + 2e 4t and y 2 = 6e 2t + 2e 4t are both known solutions of a certain equation y + p(t)y + q(t)y = g(t). (a) What is the general solution of this equation? (b) What is the equation? (That is, determine a set of functions p(t), q(t), and g(t), such that y 1 and y 2 are its solutions.) 14

15 11. Mechanical vibrations Equation at equilibrium: (Use it to find Hooke s constant.) Equation of motion: Parameters for the motion equation: mass, damping constant, Hooke s constant, applied forcing function, initial displacement and initial velocity. Equation becomes: Undamped free vibration Solution: u(t) = The displacement is a simple harmonic motion, oscillating with constant amplitude at the system s natural frequency. Know natural frequency and natural period. (What they are and how to find them.) The amplitude and phase-angle form of the displacement u(t) = Know how to find the amplitude and phase angle. Do not confuse the amplitude of the simple harmonic motion, which is constant, with amplitude of an underdamped system or a system undergoing resonance in both latter cases the amplitude is a function of time! The above formula does not apply to those latter cases. 15

16 Damped free vibration With nonzero damping constant γ, and no forcing function, there are 3 possible types of displacement function, depending on the roots of the characteristic equation. Equation: Know the 3 types of damped system, which can be classified according to the roots of the characteristic equation. Be sure to understand the differences in their behavior. Overdamped: Critically damped: Underdamped: (1) Which case(s) do not produce oscillation? (2) How often could the equilibrium position be crossed in each case? (3) Know how to find quasi frequency and quasi period. (4) As t, what happens to the displacement? (5) The amplitude of an underdamped system is not constant (as in an undamped system), but decreasing with time. How to find the maximum/peak displacement? 16

17 Ex Classify the mass-spring system described by each of the equations below as undamped, underdamped, critically damped, or overdamped. (a) u + 2u + 3u = 0 (b) 2u + 8u + 8u = 0 (c) 3u + 300u = 0 (d) 4u + 12u + 8u = 0 Ex (Exam 2, summer 2003) A mass of 1 kg stretches a spring 50 cm. The massspring system has damping of 4 N sec/m. At t = 0 the mass is set in motion from its equilibrium position with downward velocity of 2 m/sec. Assume that the gravitational constant,g = 10 m/sec 2. (a) Set up an initial value problem that describes this situation. (b) Solve the initial value problem. (c) What is the quasi-frequency of the system? 17

18 Undamped forced vibration We will only study the case of a periodic forcing function, for example F(t) = F 0 cos ωt or F(t) = F 0 sin ωt. The displacement function behaves differently depending on whether or not the forcing function s frequency is equal to the undamped system s natural frequency. What is beat? When ω = ω 0, resonance occurs. See next section. 18

19 12. Resonance A system undergoing resonance will oscillate with progressively larger amplitude that grows unbounded, increasing linearly with time (it does not grow exponentially, contrary to a popular belief). What are the precise conditions, mathematically, necessary for resonance to occur? Ex A mass-spring system is described by the equation u + 64u = 2cos(ωt) 3sin(2ωt) For what value(s) of ω will resonance occur? Ex Which equation below describes a system undergoing resonance? (a) (b) (c) (d) u + 2u + u = sin(t) u 4u = 4cos(2t) u 9u = 15sin(3t) 4u + 36u = 6cos(3t) 19

20 MATH 251 Work sheet / Things to know Chapter 4 1. Higher order linear equations Linear equations of higher ( 3) order, with constant coefficients, can be solved in the same fashion as those of the second order. For homogeneous linear equations with constant coefficients, the characteristic equation method solves them all. Know the four rules of the characteristic equation method. For an n-th order equation the general solution has exactly n terms. The characteristic equation also has exactly n roots (counting duplicates individually). Each root gives one of the n fundamental solutions, which together form the general solution, according to the four rules: i. For distinct real roots: Ex If r = 2, 4, 6, 8 are distinct roots, what are the fundamental solutions? ii. For repeated real roots: Ex If r = 6, 6, 5, 5, 5 are real roots (counting repetitions), what are the fundamental solutions? iii. For distinct complex conjugate roots: Ex If r = 2 ± 2i, 5 ± i, 1 ± 7i are distinct roots, what are the fundamental solutions? iv. For repeated complex conjugate roots: 20

21 Ex If r = 3 ± 5i, 3 ± 5i, 3 ± 5i are complex roots (counting repetitions), what are the fundamental solutions? The above 4 rules are cumulative and can be applied together. Ex Suppose an 8th order linear equation has a characteristic with the following roots: r = 3, 3, 3, 5, 4 ± i, 4 ± i. Write down its general solution. Ex Find the general solution of y (5) 16y = 0. Question: How many initial conditions must an n-th order linear equation IVP have? For nonhomogeneous linear equations of constant coefficients, the approach remains the same: use the characteristic equation to find y c, then the method of undetermined coefficients can be used to find the particular solution Y. The general solution is their sum, y = y c + Y. 21