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Appendix C Physical Constants [m0141] The speed of light in free space (c), which is the phase velocity of any electromagnetic radiation in free space, is ∼= 2.9979 × 108 m/s. This is commonly rounded up to 3 × 108 m/s. This rounding incurs error of ∼= 0.07%, which is usually much less than other errors present in elec... | Electromagnetics_Vol2_Page_225_Chunk2401 |
Index acceptance angle, 140 aluminum, 43, 46, 207 Ampere’s law general form, 7, 8, 26, 34, 105, 146, 154, 155, 159 magnetostatics, 6, 18, 124 antenna electrically-short dipole (ESD), 155–157, 159, 161, 168–172, 175–177, 198 folded half-wave dipole, 136 half-wave dipole, 136, 159, 161–162, 198 isotropic, 199, 201 micros... | Electromagnetics_Vol2_Page_226_Chunk2402 |
214 INDEX English system of units, 2 evanescent waves, see waves far field, 153, 161, 167 Faraday’s law, 7, 23 ferrite, 207 fiber optics, 88 flux electric, 7 magnetic, 6, 7 force, 20 FR4, 41, 127, 128, 136, 205 Friis transmission equation, 201–202 gain power, 39 voltage, 40 Gauss’ law electric field, 5, 18 magnetic field, 6... | Electromagnetics_Vol2_Page_227_Chunk2403 |
INDEX 215 normalized, 180 permeability, 6 of common materials, 206–207 relative, 6, 206 permittivity, 5 complex-valued, 30, 33–34 effective, 128 of common materials, 205–206 relative, 5, 205 phase velocity, 95–97, 118 in microstrip, 128 phasor, 7 plane of incidence, 70 plane wave relationships, 36, 73, 77, 107, 154, 15... | Electromagnetics_Vol2_Page_228_Chunk2404 |
216 INDEX units, 1–2 vector arithmetic, 211 identity, 211 position-free, 15 vector effective length, 183, 190, 192, 195, 196 water, 205, 207 wave equation electromagnetic, 36, 147, 149, 150 magnetic vector potential, 149, 150 source-free lossless region, 8 source-free lossy region, 30–32 wave impedance, 8, 37, 42, 154,... | Electromagnetics_Vol2_Page_229_Chunk2405 |
Electromagnetics, volume 2, by Steven W. Ellingson is a 216-page peer-reviewed open textbook designed especially for electrical engineering students in the third year of a bachelor of science degree program. It is intended as the primary textbook for the second semester of a two-semester undergraduate engineering elect... | Electromagnetics_Vol2_Page_230_Chunk2406 |
Trinity University Trinity University Digital Commons @ Trinity Digital Commons @ Trinity Faculty Authored and Edited Books & CDs 12-2013 Elementary Differential Equations with Boundary Value Problems Elementary Differential Equations with Boundary Value Problems William F. Trench Trinity University, wtrench@trinity.ed... | Elementary Differential Equations with Boundary Value Problems_Page_1_Chunk2407 |
ELEMENTARY DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS William F. Trench Andrew G. Cowles Distinguished Professor Emeritus Department of Mathematics Trinity University San Antonio, Texas, USA wtrench@trinity.edu This book has been judged to meet the evaluation criteria set by the Edi- torial Board of the Americ... | Elementary Differential Equations with Boundary Value Problems_Page_2_Chunk2408 |
Free Edition 1.01 (December 2013) This book was published previously by Brooks/Cole Thomson Learning, 2001. This free edition is made available in the hope that it will be useful as a textbook or reference. Reproduction is permitted for any valid noncommercial educational, mathematical, or scientific purpose. However, c... | Elementary Differential Equations with Boundary Value Problems_Page_3_Chunk2409 |
TO BEVERLY | Elementary Differential Equations with Boundary Value Problems_Page_4_Chunk2410 |
Contents Chapter 1 Introduction 1 1.1 Applications Leading to Differential Equations 1.2 First Order Equations 5 1.3 Direction Fields for First Order Equations 16 Chapter 2 First Order Equations 30 2.1 Linear First Order Equations 30 2.2 Separable Equations 45 2.3 Existence and Uniqueness of Solutions of Nonlinear Equa... | Elementary Differential Equations with Boundary Value Problems_Page_5_Chunk2411 |
5.5 The Method of Undetermined Coefficients II 238 5.6 Reduction of Order 248 5.7 Variation of Parameters 255 Chapter 6 Applcations of Linear Second Order Equations 268 6.1 Spring Problems I 268 6.2 Spring Problems II 279 6.3 The RLC Circuit 290 6.4 Motion Under a Central Force 296 Chapter 7 Series Solutions of Linear S... | Elementary Differential Equations with Boundary Value Problems_Page_6_Chunk2412 |
vi Contents 10.5 Constant Coefficient Homogeneous Systems II 542 10.6 Constant Coefficient Homogeneous Systems II 556 10.7 Variation of Parameters for Nonhomogeneous Linear Systems 568 Chapter 11 Boundary Value Problems and Fourier Expansions 580 11.1 Eigenvalue Problems for y′′ + λy = 0 580 11.2 Fourier Series I 586 11.... | Elementary Differential Equations with Boundary Value Problems_Page_7_Chunk2413 |
Preface Elementary Differential Equations with Boundary Value Problems is written for students in science, en- gineering, and mathematics who have completed calculus through partial differentiation. If your syllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some prepa- ra... | Elementary Differential Equations with Boundary Value Problems_Page_8_Chunk2414 |
viii Preface focuses the student’s attention on the idea of seeking a solution y of a differential equation by writing it as y = uy1, where y1 is a known solution of related equation and u is a function to be determined. I use this idea in nonstandard ways, as follows: • In Section 2.4 to solve nonlinear first order equ... | Elementary Differential Equations with Boundary Value Problems_Page_9_Chunk2415 |
Preface ix the homogeneous boundary conditions. Similarly, in most of the examples and exercises Section 12.3 (Laplace’s Equation), the functions defining the boundary conditions on a given side of the rectangular domain satisfy homogeneous boundary conditions at the endpoints of the same type (Dirichlet or Neu- mann) a... | Elementary Differential Equations with Boundary Value Problems_Page_10_Chunk2416 |
CHAPTER 1 Introduction IN THIS CHAPTER we begin our study of differential equations. SECTION 1.1 presents examples of applications that lead to differential equations. SECTION 1.2 introduces basic concepts and definitions concerning differential equations. SECTION 1.3 presents a geometric method for dealing with differe... | Elementary Differential Equations with Boundary Value Problems_Page_11_Chunk2417 |
2 Chapter 1 Introduction 1.1 APPLICATIONS LEADING TO DIFFERENTIAL EQUATIONS In order to apply mathematical methods to a physical or “real life” problem, we must formulate the prob- lem in mathematical terms; that is, we must construct a mathematical model for the problem. Many physical problems concern relationships be... | Elementary Differential Equations with Boundary Value Problems_Page_12_Chunk2418 |
Section 1.1 Applications Leading to Differential Equations 3 (When you see a name in blue italics, just click on it for information about the person.) This model assumes that the numbers of births and deaths per unit time are both proportional to the population. The constants of proportionality are the birth rate (birt... | Elementary Differential Equations with Boundary Value Problems_Page_13_Chunk2419 |
4 Chapter 1 Introduction P t 1/α Figure 1.1.1 Solutions of the logistic equation where T0 is the temperature of the body when t = 0. Therefore limt→∞T(t) = Tm, independent of T0. (Common sense suggests this. Why?) Figure 1.1.2 shows typical graphs of T versus t for various values of T0. Assuming that the medium remains... | Elementary Differential Equations with Boundary Value Problems_Page_14_Chunk2420 |
Section 1.1 Applications Leading to Differential Equations 5 T t t Tm Figure 1.1.2 Temperature according to Newton’s Law of Cooling Glucose Absorption by the Body Glucose is absorbed by the body at a rate proportionalto the amount of glucose present in the bloodstream. Let λ denote the (positive) constant of proportion... | Elementary Differential Equations with Boundary Value Problems_Page_15_Chunk2421 |
6 Chapter 1 Introduction Graphs of this function are similar to those in Figure 1.1.2. (Why?) Spread of Epidemics One model for the spread of epidemics assumes that the number of people infected changes at a rate proportional to the product of the number of people already infected and the number of people who are susce... | Elementary Differential Equations with Boundary Value Problems_Page_16_Chunk2422 |
Section 1.2 Basic Concepts 7 where α and β are positive constants. (Since negative population doesn’t make sense, this system works only while P and Q are both positive.) Now suppose P (0) = P0 > 0 and Q(0) = Q0 > 0. It can be shown (Exercise 10.4.42) that there’s a positive constant ρ such that if (P0, Q0) is above th... | Elementary Differential Equations with Boundary Value Problems_Page_17_Chunk2423 |
8 Chapter 1 Introduction then y = Z x3 dx = x4 4 + c, where c is an arbitrary constant. If n > 1 we can find functions y that satisfy equations of the form y(n) = f(x) (1.2.1) by repeated integration. Again, this is a calculus problem. Except for illustrative purposes in this section, there’s no need to consider differe... | Elementary Differential Equations with Boundary Value Problems_Page_18_Chunk2424 |
Section 1.2 Basic Concepts 9 Example 1.2.1 If a is any positive constant, the circle x2 + y2 = a2 (1.2.3) is an integral curve of y′ = −x y . (1.2.4) To see this, note that the only functions whose graphs are segments of (1.2.3) are y1 = p a2 −x2 and y2 = − p a2 −x2. We leave it to you to verify that these functions bo... | Elementary Differential Equations with Boundary Value Problems_Page_19_Chunk2425 |
10 Chapter 1 Introduction x y 0.5 1.0 1.5 2.0 −0.5 −1.0 −1.5 −2.0 2 4 6 8 −2 −4 −6 −8 Figure 1.2.1 y = x2 3 + 1 x so y′′ + 2y′ + y = (c1 + c2x)e−x −2c2e−x +2 −(c1 + c2x)e−x + c2e−x + 2 +(c1 + c2x)e−x + 2x −4 = (1 −2 + 1)(c1 + c2x)e−x + (−2 + 2)c2e−x +4 + 2x −4 = 2x for all values of x. Therefore y is a solution of (1... | Elementary Differential Equations with Boundary Value Problems_Page_20_Chunk2426 |
Section 1.2 Basic Concepts 11 where k1, k2, ..., kn are constants. This shows that every solution of (1.2.9) has the form (1.2.10) for some choice of the constants k1, k2, ..., kn. On the other hand, differentiating (1.2.10) n times shows that if k1, k2, ..., kn are arbitrary constants, then the function y in (1.2.10) ... | Elementary Differential Equations with Boundary Value Problems_Page_21_Chunk2427 |
12 Chapter 1 Introduction is an initial value problem for a second order differential equation where y and y′ are required to have specified values at x = 0. In general, an initial value problem for an n-th order differential equation requires y and its first n−1 derivatives to have specified values at some point x0. Thes... | Elementary Differential Equations with Boundary Value Problems_Page_22_Chunk2428 |
Section 1.2 Basic Concepts 13 Example 1.2.7 In Example 1.2.2 we verified that y = x2 3 + 1 x (1.2.14) is a solution of xy′ + y = x2 on (0, ∞) and on (−∞, 0). By evaluating (1.2.14) at x = ±1, you can see that (1.2.14) is a solution of the initial value problems xy′ + y = x2, y(1) = 4 3 (1.2.15) and xy′ + y = x2, y(−1) =... | Elementary Differential Equations with Boundary Value Problems_Page_23_Chunk2429 |
14 Chapter 1 Introduction 1.2 Exercises 1. Find the order of the equation. (a) d2y dx2 + 2 dy dx d3y dx3 + x = 0 (b) y′′ −3y′ + 2y = x7 (c) y′ −y7 = 0 (d) y′′y −(y′)2 = 2 2. Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the f... | Elementary Differential Equations with Boundary Value Problems_Page_24_Chunk2430 |
Section 1.2 Basic Concepts 15 (c) y = tan x2 2 ; y′ = x(1 + y2), y(0) = 0 (d) y = 2 x −2; y′ = −y(y + 1) x , y(1) = −2 6. Verify that the function is a solution of the initial value problem. (a) y = x2(1 + ln x); y′′ = 3xy′ −4y x2 , y(e) = 2e2, y′(e) = 5e (b) y = x2 3 + x −1; y′′ = x2 −xy′ + y + 1 x2 , y(1) = 1 3, y... | Elementary Differential Equations with Boundary Value Problems_Page_25_Chunk2431 |
16 Chapter 1 Introduction 1.3 DIRECTION FIELDS FOR FIRST ORDER EQUATIONS It’s impossible to find explicit formulas for solutions of some differential equations. Even if there are such formulas, they may be so complicated that they’re useless. In this case we may resort to graphical or numerical methods to get some idea ... | Elementary Differential Equations with Boundary Value Problems_Page_26_Chunk2432 |
Section 1.3 Direction Fields for First Order Equations 17 y x a b c d Figure 1.3.1 A rectangular grid Unfortunately, approximating a direction field and graphing integral curves in this way is too tedious to be done effectively by hand. However, there is software for doing this. As you’ll see, the combina- tion of direc... | Elementary Differential Equations with Boundary Value Problems_Page_27_Chunk2433 |
18 Chapter 1 Introduction −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y x Figure 1.3.4 A direction and integral curves for y′ = x −y 1 + x2 The methods of Chapter 3 won’t work for the equation y′ = −x/y (1.3.2) if R contains part of the x-axis, since f(x, y) = −x/y is undefined ... | Elementary Differential Equations with Boundary Value Problems_Page_28_Chunk2434 |
Section 1.3 Direction Fields for First Order Equations 19 Eqns. (1.3.2) and (1.3.3) can be reformulated as in (1.3.4) with dx dt = −y, dy dt = x and dx dt = 1 −x2 −y2, dy dt = x2, respectively. Even if f is continuous and otherwise “nice” throughout R, your software may require you to reformulate the equation y′ = f(x,... | Elementary Differential Equations with Boundary Value Problems_Page_29_Chunk2435 |
20 Chapter 1 Introduction As you study from this book, you’ll often be asked to use computer software and graphics. Exercises with this intent are marked as C (computer or calculator required), C/G (computer and/or graphics required), or L (laboratory work requiring software and/or graphics). Often you may not complete... | Elementary Differential Equations with Boundary Value Problems_Page_30_Chunk2436 |
Section 1.3 Direction Fields for First Order Equations 21 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 1 A direction field for y′ = x y | Elementary Differential Equations with Boundary Value Problems_Page_31_Chunk2437 |
22 Chapter 1 Introduction 0 0.5 1 1.5 2 2.5 3 3.5 4 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y 2 A direction field for y′ = 2xy2 1 + x2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 3 A direction field for y′ = x2(1 + y2) | Elementary Differential Equations with Boundary Value Problems_Page_32_Chunk2438 |
Section 1.3 Direction Fields for First Order Equations 23 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 4 A direction field for y′ = 1 1 + x2 + y2 0 0.5 1 1.5 2 2.5 3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 5 A direction field for y′ = −(2xy2 + y3) | Elementary Differential Equations with Boundary Value Problems_Page_33_Chunk2439 |
24 Chapter 1 Introduction −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 6 A direction field for y′ = (x2 + y2)1/2 0 1 2 3 4 5 6 7 −3 −2 −1 0 1 2 3 x y 7 A direction field for y′ = sin xy | Elementary Differential Equations with Boundary Value Problems_Page_34_Chunk2440 |
Section 1.3 Direction Fields for First Order Equations 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y 8 A direction field for y′ = exy 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 9 A direction field for y′ = (x −y2)(x2 −y) | Elementary Differential Equations with Boundary Value Problems_Page_35_Chunk2441 |
26 Chapter 1 Introduction 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y 10 A direction field for y′ = x3y2 + xy3 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 x y 11 A direction field for y′ = sin(x −2y) | Elementary Differential Equations with Boundary Value Problems_Page_36_Chunk2442 |
Section 1.3 Direction Fields for First Order Equations 27 In Exercises 12-22 construct a direction field and plot some integral curves in the indicated rectangular region. 12. C/G y′ = y(y −1); {−1 ≤x ≤2, −2 ≤y ≤2} 13. C/G y′ = 2 −3xy; {−1 ≤x ≤4, −4 ≤y ≤4} 14. C/G y′ = xy(y −1); {−2 ≤x ≤2, −4 ≤y ≤4} 15. C/G y′ = 3x + y;... | Elementary Differential Equations with Boundary Value Problems_Page_37_Chunk2443 |
28 Chapter 1 Introduction discussed in Section 1.1 in connection with Verhulst’s population model and the spread of an epidemic, we can write both in the form y′ = ay −by2, (C) where a and b are positive constants. Thus, (A) is of the form (C) with y = P , a = a, and b = aα, and (B) is of the form (C) with y = I, a = r... | Elementary Differential Equations with Boundary Value Problems_Page_38_Chunk2444 |
CHAPTER 2 First Order Equations IN THIS CHAPTER we study first order equations for which there are general methods of solution. SECTION 2.1 deals with linear equations, the simplest kind of first order equations. In this section we introduce the method of variation of parameters. The idea underlying this method will be a... | Elementary Differential Equations with Boundary Value Problems_Page_39_Chunk2445 |
30 Chapter 2 First Order Equations 2.1 LINEAR FIRST ORDER EQUATIONS A first order differential equation is said to be linear if it can be written as y′ + p(x)y = f(x). (2.1.1) A first order differential equation that can’t be written like this is nonlinear. We say that (2.1.1) is homogeneous if f ≡0; otherwise it’s nonho... | Elementary Differential Equations with Boundary Value Problems_Page_40_Chunk2446 |
Section 2.1 Linear First Order Equations 31 (−∞, 0) and (0, ∞); moreover, every solution of (2.1.2) on either of these intervals is of the form (2.1.3) for some choice of c. We say that (2.1.3) is the general solution of (2.1.2). We’ll see that a similar situation occurs in connection with any first order linear equatio... | Elementary Differential Equations with Boundary Value Problems_Page_41_Chunk2447 |
32 Chapter 2 First Order Equations x 0.2 0.4 0.6 0.8 1.0 y 0.5 1.0 1.5 2.0 2.5 3.0 a = 2 a = 1.5 a = 1 a = −1 a = −2.5 a = −4 Figure 2.1.1 Solutions of y′ −ay = 0, y(0) = 1 for x in I. Integrating this shows that ln|y| = ax + k, so |y| = ekeax, where k is an arbitrary constant. Since eax can never equal zero, y has no ... | Elementary Differential Equations with Boundary Value Problems_Page_42_Chunk2448 |
Section 2.1 Linear First Order Equations 33 SOLUTION(a) We rewrite (2.1.8) as y′ + 1 xy = 0, (2.1.10) where x is restricted to either (−∞, 0) or (0, ∞). If y is a nontrivial solution of (2.1.10), there must be some open interval I on which y has no zeros. We can rewrite (2.1.10) as y′ y = −1 x for x in I. Integrating s... | Elementary Differential Equations with Boundary Value Problems_Page_43_Chunk2449 |
34 Chapter 2 First Order Equations x y c > 0 c < 0 c > 0 c < 0 Figure 2.1.2 Solutions of xy′ + y = 0 on (0, ∞) and (−∞, 0) Proof If y = ce−P(x), differentiating y and using (2.1.14) shows that y′ = −P ′(x)ce−P(x) = −p(x)ce−P(x) = −p(x)y, so y′ + p(x)y = 0; that is, y is a solution of (2.1.12), for any choice of c. Now ... | Elementary Differential Equations with Boundary Value Problems_Page_44_Chunk2450 |
Section 2.1 Linear First Order Equations 35 Linear Nonhomogeneous First Order Equations We’ll now solve the nonhomogeneous equation y′ + p(x)y = f(x). (2.1.16) When considering this equation we call y′ + p(x)y = 0 the complementary equation. We’ll find solutions of (2.1.16) in the form y = uy1, where y1 is a nontrivial ... | Elementary Differential Equations with Boundary Value Problems_Page_45_Chunk2451 |
36 Chapter 2 First Order Equations −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y Figure 2.1.3 A direction field and integral curves for y′ + 2y = x2e−2x and y = ue−2x = e−2x x4 4 + c is the general solution of (2.1.18). Figure 2.1.3 shows a direction field and some integral curves for (2... | Elementary Differential Equations with Boundary Value Problems_Page_46_Chunk2452 |
Section 2.1 Linear First Order Equations 37 so that y′ = u′ sin x −u cos x sin2 x (2.1.23) and y′ + (cot x)y = u′ sin x −u cos x sin2 x + u cot x sin x = u′ sin x −u cos x sin2 x + u cos x sin2 x = u′ sin x. (2.1.24) Therefore y is a solution of (2.1.20) if and only if u′/ sin x = x csc x = x/ sinx or, equivalently, u′... | Elementary Differential Equations with Boundary Value Problems_Page_47_Chunk2453 |
38 Chapter 2 First Order Equations REMARK: It wasn’t necessary to do the computations (2.1.23) and (2.1.24) in Example 2.1.6, since we showed in the discussion preceding Example 2.1.5 that if y = uy1 where y′ 1 + p(x)y1 = 0, then y′ + p(x)y = u′y1. We did these computations so you would see this happen in this specific ... | Elementary Differential Equations with Boundary Value Problems_Page_48_Chunk2454 |
Section 2.1 Linear First Order Equations 39 Integrating this and taking the constant of integration to be zero yields ln|y1| = x2, so |y1| = ex2. We choose y1 = ex2 and seek solutions of (2.1.28) in the form y = uex2, where u′ex2 = 1, so u′ = e−x2. Therefore u = c + Z e−x2dx, but we can’t simplify the integral on the r... | Elementary Differential Equations with Boundary Value Problems_Page_49_Chunk2455 |
40 Chapter 2 First Order Equations An Existence and Uniqueness Theorem The method of variation of parameters leads to this theorem. Theorem 2.1.2 Suppose p and f are continuous on an open interval (a, b), and let y1 be any nontrivial solution of the complementary equation y′ + p(x)y = 0 on (a, b). Then: (a) The general... | Elementary Differential Equations with Boundary Value Problems_Page_50_Chunk2456 |
Section 2.1 Linear First Order Equations 41 Integrating u′ = f/y1 yields u = c + Z f(x)/y1(x) dx , which implies (2.1.32), since y = uy1. (b) We’ve proved (a), where R f(x)/y1(x) dx in (2.1.32) is an arbitrary antiderivative of f/y1. Now it’s convenient to choose the antiderivative that equals zero when x = x0, and... | Elementary Differential Equations with Boundary Value Problems_Page_51_Chunk2457 |
42 Chapter 2 First Order Equations In Exercises 16 –24 find the general solution. 16. y′ + 1 xy = 7 x2 + 3 17. y′ + 4 x −1y = 1 (x −1)5 + sinx (x −1)4 18. xy′ + (1 + 2x2)y = x3e−x2 19. xy′ + 2y = 2 x2 + 1 20. y′ + (tan x)y = cos x 21. (1 + x)y′ + 2y = sin x 1 + x 22. (x −2)(x −1)y′ −(4x −3)y = (x −2)3 23. y′ + (2 sinx c... | Elementary Differential Equations with Boundary Value Problems_Page_52_Chunk2458 |
Section 2.1 Linear First Order Equations 43 41. y′ + 2x 1 + x2 y = ex (1 + x2)2 , y(0) = 1 42. xy′ + (x + 1)y = ex2, y(1) = 2 43. Experiments indicate that glucose is absorbed by the body at a rate proportional to the amount of glucose present in the bloodstream. Let λ denote the (positive) constant of proportionality.... | Elementary Differential Equations with Boundary Value Problems_Page_53_Chunk2459 |
44 Chapter 2 First Order Equations 46. Assume that all functions in this exercise are defined on a common interval (a, b). (a) Prove: If y1 and y2 are solutions of y′ + p(x)y = f1(x) and y′ + p(x)y = f2(x) respectively, and c1 and c2 are constants, then y = c1y1 + c2y2 is a solution of y′ + p(x)y = c1f1(x) + c2f2(x). (T... | Elementary Differential Equations with Boundary Value Problems_Page_54_Chunk2460 |
Section 2.2 Separable Equations 45 2.2 SEPARABLE EQUATIONS A first order differential equation is separable if it can be written as h(y)y′ = g(x), (2.2.1) where the left side is a product of y′ and a function of y and the right side is a function of x. Rewriting a separable differential equation in this form is called s... | Elementary Differential Equations with Boundary Value Problems_Page_55_Chunk2461 |
46 Chapter 2 First Order Equations Example 2.2.2 (a) Solve the equation y′ = −x y . (2.2.4) (b) Solve the initial value problem y′ = −x y , y(1) = 1. (2.2.5) (c) Solve the initial value problem y′ = −x y , y(1) = −2. (2.2.6) SOLUTION(a) Separating variables in (2.2.4) yields yy′ = −x. Integrating yields y2 2 = −x2 2 + ... | Elementary Differential Equations with Boundary Value Problems_Page_56_Chunk2462 |
Section 2.2 Separable Equations 47 x y 1 2 −1 −2 1 2 −1 −2 (a) (b) Figure 2.2.1 (a) y = √ 2 −x2, − √ 2 < x < √ 2; (b) y = − √ 5 −x2, − √ 5 < x < √ 5 Implicit Solutions of Separable Equations In Examples 2.2.1 and 2.2.2 we were able to solve the equation H(y) = G(x) + c to obtain explicit formulas for solutions of the g... | Elementary Differential Equations with Boundary Value Problems_Page_57_Chunk2463 |
48 Chapter 2 First Order Equations • The function y in (2.2.11) (not (2.2.11) itself) is a solution of h(y)y′ = g(x). Example 2.2.3 (a) Find implicit solutions of y′ = 2x + 1 5y4 + 1. (2.2.13) (b) Find an implicit solution of y′ = 2x + 1 5y4 + 1, y(2) = 1. (2.2.14) SOLUTION(a) Separating variables yields (5y4 + 1)y′ = ... | Elementary Differential Equations with Boundary Value Problems_Page_58_Chunk2464 |
Section 2.2 Separable Equations 49 1 1.5 2 2.5 3 3.5 4 −1 −0.5 0 0.5 1 1.5 2 x y Figure 2.2.2 A direction field and integral curves for y′ = 2x + 1 5y4 + 1 Integrating this yields −1 y = x2 + c, which is equivalent to y = − 1 x2 + c. (2.2.17) We’ve now shown that if y is a solution of (2.2.16) that is not identically ze... | Elementary Differential Equations with Boundary Value Problems_Page_59_Chunk2465 |
50 Chapter 2 First Order Equations −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 y x Figure 2.2.3 A direction field and integral curves for y′ = 2xy2 A partial fraction expansion on the left yields 1 y −1 − 1 y + 1 y′ = −x, and integrating yields ln y −1 y + 1 = −x2 2 + k; hence, y −1 y + 1 = eke−x2/2.... | Elementary Differential Equations with Boundary Value Problems_Page_60_Chunk2466 |
Section 2.2 Separable Equations 51 constant solution y ≡1 can be obtained from this formula by taking c = 0; however, the other constant solution, y ≡−1, can’t be obtained in this way. Figure 2.2.4 shows a direction field and some integrals for (2.2.18). −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −2 −1 0 1 2 3 x y Figure 2.2.4 A ... | Elementary Differential Equations with Boundary Value Problems_Page_61_Chunk2467 |
52 Chapter 2 First Order Equations Example 2.2.6 Solve the initial value problem y′ = 2xy2, y(0) = y0 and determine the interval of validity of the solution. Solution First suppose y0 ̸= 0. From Example 2.2.4, we know that y must be of the form y = − 1 x2 + c. (2.2.20) Imposing the initial condition shows that c = −1/y... | Elementary Differential Equations with Boundary Value Problems_Page_62_Chunk2468 |
Section 2.2 Separable Equations 53 14. C/G y′ + (y + 1)(y −1)(y −2) x + 1 = 0, y(1) = 0 15. C/G y′ + 2x(y + 1) = 0, y(0) = 2 16. C/G y′ = 2xy(1 + y2), y(0) = 1 In Exercises 17–23 solve the initial value problem and find the interval of validity of the solution. 17. y′(x2 + 2) + 4x(y2 + 2y + 1) = 0, y(1) = −1 18. y′ = −2... | Elementary Differential Equations with Boundary Value Problems_Page_63_Chunk2469 |
54 Chapter 2 First Order Equations (a) Choose r and S positive. By plotting direction fields and solutions of (A) on suitable rectan- gular grids R = {0 ≤t ≤T, 0 ≤I ≤d} in the (t, I)-plane, verify that if I is any solution of (A) such that I(0) > 0, then limt→∞I(t) = S −q/r if q < rS and limt→∞I(t) = 0 if q ≥rS. (b) To ... | Elementary Differential Equations with Boundary Value Problems_Page_64_Chunk2470 |
Section 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 55 34. Assuming that p ̸≡0, state conditions under which the linear equation y′ + p(x)y = f(x) is separable. If the equation satisfies these conditions, solve it by separation of variables and by the method developed in Section 2.1. Solve the equat... | Elementary Differential Equations with Boundary Value Problems_Page_65_Chunk2471 |
56 Chapter 2 First Order Equations Some terminology: an open rectangle R is a set of points (x, y) such that a < x < b and c < y < d (Figure 2.3.1). We’ll denote this set by R : {a < x < b, c < y < d}. “Open” means that the boundary rectangle (indicated by the dashed lines in Figure 2.3.1) isn’t included in R . The nex... | Elementary Differential Equations with Boundary Value Problems_Page_66_Chunk2472 |
Section 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 57 are continuous for all (x, y), Theorem 2.3.1 implies that if (x0, y0) is arbitrary, then (2.3.3) has a unique solution on some open interval that contains x0. Example 2.3.2 Consider the initial value problem y′ = x2 −y2 x2 + y2 , y(x0) = y0. (2... | Elementary Differential Equations with Boundary Value Problems_Page_67_Chunk2473 |
58 Chapter 2 First Order Equations Example 2.3.5 Consider the initial value problem y′ = 10 3 xy2/5, y(x0) = y0. (2.3.8) (a) For what points (x0, y0) does Theorem 2.3.1(a) imply that (2.3.8) has a solution? (b) For what points (x0, y0) does Theorem 2.3.1(b) imply that (2.3.8) has a unique solution on some open interval... | Elementary Differential Equations with Boundary Value Problems_Page_68_Chunk2474 |
Section 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 59 x y Figure 2.3.2 Two solutions (y = 0 and y = x1/2) of (2.3.9) that differ on every interval containing x0 = 0 Therefore (2.3.11) satisfies (2.3.10) on (−∞, ∞) even if c ≤0, so that y( p |c|) = y(− p |c|) = 0. In particular, taking c = 0 in (2.3... | Elementary Differential Equations with Boundary Value Problems_Page_69_Chunk2475 |
60 Chapter 2 First Order Equations Exercise 2.2.15, there are infinitely many other solutions of (2.3.12) that differ from (2.3.13) on every open interval larger than (−1, 1). One such solution is y = ( (x2 −1)5/3, −1 ≤x ≤1, 0, |x| > 1. (Figure 2.3.3). 1 −1 x y (0, −1) Figure 2.3.3 Two solutions of (2.3.12) on (−∞, ∞) t... | Elementary Differential Equations with Boundary Value Problems_Page_70_Chunk2476 |
Section 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 61 1. y′ = x2 + y2 sin x 2. y′ = ex + y x2 + y2 3. y′ = tan xy 4. y′ = x2 + y2 ln xy 5. y′ = (x2 + y2)y1/3 6. y′ = 2xy 7. y′ = ln(1 + x2 + y2) 8. y′ = 2x + 3y x −4y 9. y′ = (x2 + y2)1/2 10. y′ = x(y2 −1)2/3 11. y′ = (x2 + y2)2 12. y′ = (x + y)1/2 ... | Elementary Differential Equations with Boundary Value Problems_Page_71_Chunk2477 |
62 Chapter 2 First Order Equations 16. Use the ideas developed in Exercise 15 to find infinitely many solutionsof the initial value problem y′ = y2/5, y(0) = 1 on (−∞, ∞). 17. Consider the initial value problem y′ = 3x(y −1)1/3, y(x0) = y0. (A) (a) For what points (x0, y0) does Theorem 2.3.1 imply that (A) has a solution... | Elementary Differential Equations with Boundary Value Problems_Page_72_Chunk2478 |
Section 2.4 Transformation of Nonlinear Equations into Separable Equations 63 are of the form y = uy1, where y1 is a nontrivial solution of the complementary equation y′ + p(x)y = 0 (2.4.1) and u is a solution of u′y1(x) = f(x). Note that this last equation is separable, since it can be rewritten as u′ = f(x) y1(x). In... | Elementary Differential Equations with Boundary Value Problems_Page_73_Chunk2479 |
64 Chapter 2 First Order Equations −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y Figure 2.4.1 A direction field and integral curves for y′ −y = xy2 and y = − 1 x −1 + ce−x . Figure 2.4.1 shows direction field and some integral curves of (2.4.3). Other Nonlinear Equations That Can be Transformed Into Sep... | Elementary Differential Equations with Boundary Value Problems_Page_74_Chunk2480 |
Section 2.4 Transformation of Nonlinear Equations into Separable Equations 65 Homogeneous Nonlinear Equations In the text we’ll consider only the most widely studied class of equations for which the method of the preceding paragraph works. Other types of equations appear in Exercises 44–51. The differential equation (2... | Elementary Differential Equations with Boundary Value Problems_Page_75_Chunk2481 |
66 Chapter 2 First Order Equations Integrating yields eu = ln |x| + c. Therefore u = ln(ln |x| + c) and y = ux = x ln(ln |x| + c). Figure 2.4.2 shows a direction field and integral curves for (2.4.8). 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y Figure 2.4.2 A direction field and some integral curves f... | Elementary Differential Equations with Boundary Value Problems_Page_76_Chunk2482 |
Section 2.4 Transformation of Nonlinear Equations into Separable Equations 67 By inspection this equation has the constant solutions u ≡1 and u ≡−1. Therefore y = x and y = −x are solutions of (2.4.9). If u is a solution of (2.4.11) that doesn’t assume the values ±1 on some interval, separating variables yields u′ u2 −... | Elementary Differential Equations with Boundary Value Problems_Page_77_Chunk2483 |
68 Chapter 2 First Order Equations The situation is more complicated if x = 0 is the open interval. First, note that y = −x satisfies (2.4.9) on (−∞, ∞). If c1 and c2 are arbitrary constants, the function y = x(1 + c1x2) 1 −c1x2 , a < x < 0, x(1 + c2x2) 1 −c2x2 , 0 ≤x < b, (2.4.14) is a solution of (2.4.9)... | Elementary Differential Equations with Boundary Value Problems_Page_78_Chunk2484 |
Section 2.4 Transformation of Nonlinear Equations into Separable Equations 69 5. C/G y′ −xy = x3y3; {−3 ≤x ≤3, 2 ≤y ≥2} 6. C/G y′ −1 + x 3x y = y4; {−2 ≤x ≤2, −2 ≤y ≤2} In Exercises 7–11 solve the initial value problem. 7. y′ −2y = xy3, y(0) = 2 √ 2 8. y′ −xy = xy3/2, y(1) = 4 9. xy′ + y = x4y4, y(1) = 1/2 10. y′ −2y =... | Elementary Differential Equations with Boundary Value Problems_Page_79_Chunk2485 |
70 Chapter 2 First Order Equations In Exercises 19-21 solve the equation explicitly. Also, plot a direction field and some integral curves on the indicated rectangular region. 19. C/G x2y′ = xy + x2 + y2; {−8 ≤x ≤8, −8 ≤y ≤8} 20. C/G xyy′ = x2 + 2y2; {−4 ≤x ≤4, −4 ≤y ≤4} 21. C/G y′ = 2y2 + x2e−(y/x)2 2xy ; {−8 ≤x ≤8, −8... | Elementary Differential Equations with Boundary Value Problems_Page_80_Chunk2486 |
Section 2.4 Transformation of Nonlinear Equations into Separable Equations 71 (e) Graph other solutions of (A) that are defined only on intervals of the form (−∞, a), where is a finite positive number. 36. L (a) Solve the equation xyy′ = x2 −xy + y2 (A) implicitly. (b) Plot a direction field for (A) on a square {0 ≤x ≤r, ... | Elementary Differential Equations with Boundary Value Problems_Page_81_Chunk2487 |
72 Chapter 2 First Order Equations 40. Prove: If ad −bc ̸= 0, the equation y′ = ax + by + α cx + dy + β can be transformed into the homogeneous nonlinear equation dY dX = aX + bY cX + dY by the substitution x = X −X0, y = Y −Y0, where X0 and Y0 are suitably chosen constants. In Exercises 41-43 use a method suggested by... | Elementary Differential Equations with Boundary Value Problems_Page_82_Chunk2488 |
Section 2.5 Exact Equations 73 In Exercises 56–59, given that y1 is a solution of the given equation, use the method suggested by Exercise 55 to find other solutions. 56. y′ = 1 + x −(1 + 2x)y + xy2; y1 = 1 57. y′ = e2x + (1 −2ex)y + y2; y1 = ex 58. xy′ = 2 −x + (2x −2)y −xy2; y1 = 1 59. xy′ = x3 + (1 −2x2)y + xy2; y1 =... | Elementary Differential Equations with Boundary Value Problems_Page_83_Chunk2489 |
74 Chapter 2 First Order Equations Example 2.5.1 Show that x4y3 + x2y5 + 2xy = c (2.5.4) is an implicit solution of (4x3y3 + 2xy5 + 2y) dx + (3x4y2 + 5x2y4 + 2x) dy = 0. (2.5.5) Solution Regarding y as a function of x and differentiating (2.5.4) implicitly with respect to x yields (4x3y3 + 2xy5 + 2y) + (3x4y2 + 5x2y4 +... | Elementary Differential Equations with Boundary Value Problems_Page_84_Chunk2490 |
Section 2.5 Exact Equations 75 QUESTION 1. Given an equation (2.5.8), how can we determine whether it’s exact? QUESTION 2. If (2.5.8) is exact, how do we find a function F satisfying (2.5.9)? To discover the answer to Question 1, assume that there’s a function F that satisfies (2.5.9) on some open rectangle R, and in add... | Elementary Differential Equations with Boundary Value Problems_Page_85_Chunk2491 |
76 Chapter 2 First Order Equations and My(x, y) = Nx(x, y) = 12x3y2 for all (x, y). Therefore Theorem 2.5.2 implies that there’s a function F such that Fx(x, y) = M(x, y) = 4x3y3 + 3x2 (2.5.14) and Fy(x, y) = N(x, y) = 3x4y2 + 6y2 (2.5.15) for all (x, y). To find F , we integrate (2.5.14) with respect to x to obtain F (... | Elementary Differential Equations with Boundary Value Problems_Page_86_Chunk2492 |
Section 2.5 Exact Equations 77 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y x Figure 2.5.1 A direction field and integral curves for (4x3y3 + 3x2) dx + (3x4y2 + 6y2) dy = 0 Substituting this into (2.5.18) yields (2.5.17). Figure 2.5.1 shows a direction field and some integral cu... | Elementary Differential Equations with Boundary Value Problems_Page_87_Chunk2493 |
78 Chapter 2 First Order Equations Step 5. Integrate φ′ with respect to y, taking the constant of integration to be zero, and substitute the result in (2.5.20) to obtain F (x, y). Step 6. Set F (x, y) = c to obtain an implicit solution of (2.5.19). If possible, solve for y explicitly as a function of x. It’s a common m... | Elementary Differential Equations with Boundary Value Problems_Page_88_Chunk2494 |
Section 2.5 Exact Equations 79 Solution Here My(x, y) = 6x2y and Nx(x, y) = 18x2y, so (2.5.25) isn’t exact. Nevertheless, let’s try to find a function F such that Fx(x, y) = 3x2y2 (2.5.26) and Fy(x, y) = 6x3y. (2.5.27) Integrating (2.5.26) with respect to x yields F (x, y) = x3y2 + φ(y), and differentiating this with re... | Elementary Differential Equations with Boundary Value Problems_Page_89_Chunk2495 |
80 Chapter 2 First Order Equations 16. | Elementary Differential Equations with Boundary Value Problems_Page_90_Chunk2496 |
Section 2.5 Exact Equations 81 (c) Plot a direction field and some integral curves for (A) on a rectangular region centered at the origin. What is the interval of validity of the solution of (B)? 28. L (a) Solve the exact equation (x2 + y2) dx + 2xy dy = 0 (A) implicitly. (b) For what choices of (x0, y0) does Theorem 2.... | Elementary Differential Equations with Boundary Value Problems_Page_91_Chunk2497 |
82 Chapter 2 First Order Equations 35. Suppose M and N are continuous and have continuous partial derivatives My and Nx that satisfy the exactness condition My = Nx on an open rectangle R. Show that if (x, y) is in R and F (x, y) = Z x x0 M(s, y0) ds + Z y y0 N(x, t) dt, then Fx = M and Fy = N. 36. Under the assumption... | Elementary Differential Equations with Boundary Value Problems_Page_92_Chunk2498 |
Section 2.6 Exact Equations 83 44. Verify that the following functions are harmonic, and find all their harmonic conjugates. (See Exercise 43.) (a) x2 −y2 (b) ex cos y (c) x3 −3xy2 (d) cos x cosh y (e) sin x cosh y 2.6 INTEGRATING FACTORS In Section 2.5 we saw that if M, N, My and Nx are continuous and My = Nx on an ope... | Elementary Differential Equations with Boundary Value Problems_Page_93_Chunk2499 |
84 Chapter 2 Integrating Factors function of y; that is, µ(x, y) = P (x)Q(y). We’re not saying that every equation M dx + N dy = 0 has an integrating factor of this form; rather, we’re saying that some equations have such integrating factors.We’llnow develop a way to determine whether a given equation has such an integ... | Elementary Differential Equations with Boundary Value Problems_Page_94_Chunk2500 |
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