(常微分方程式)

1、Textbook Advanced Engineering Mathematics, 9th Edition by Erwin Kreyszig, ISBN 0-470-07446-9, 2006.Advanced Mathematics may include the following topics:1. Ordinary Differential Equations (常微分方程式) ()2. Vector Analysis (向量分析) ()3. Fourier Analysis (傅立葉分析) ()4. Linear Algebra (線性代數) (；選修)5. Numerical

2、Analysis (數值分析) (；選修)6. Complex Analysis (複變分析) () 7. Partial Differential Equations (偏微分方程式) ()Part A Ordinary Differential Equations (常微分方程式：內含單變數函數的導數)Differential equations are of basic importance in engineering mathematics because many physical laws and relations appear mathematically in the fo

3、rm of a differential equation.Fig. 1 Applications of differential equations (微分方程式的應用)Chapter 1 First-Order ODEs Chapter 2 Second-Order Linear ODEs Chapter 3 Higher-Order Linear ODEs Chapter 5 Series Solutions of ODEs Chapter 6 Laplace Transforms1.1 Basic ConceptsIf you want to solve an engineering

4、problem (usually of a physical nature), we first have to formulate the problem as a mathematical expression in terms of variables, functions, equations, and so forth. Such an expression is known as a mathematical model of the given problem. (微分方程式是對應的工程問題的一種數學模型)The process of setting up a model, so

5、lving it mathematically, and interpreting the result in physical or other terms is called mathematical modeling (數學模型化) or, briefly, modeling (模型化).An ordinary differential equation (ODE) (常微分方程式) is an equation that contains one or several derivatives of an unknown function, which we usually call o

6、r . For example,.The term ordinary distinguishes them from partial differential equations (PDEs) (偏微分方程式), which involve partial derivatives of an unknown function of two or more variables (內含多變數函數的偏導數). For instance,.PDEs are more complicated than ODEs. (偏微分方程式較常微分方程式來得複雜及難解)An ODE is said to be of

7、 order n (n階) if the nth derivative of the unknown function y is the highest derivative of y in the equation (方程式中所出現的最高階導數為n階).The concept of order gives a useful classification into ODEs of first order, second order, and so on. (通常，階數越高越難求解，故微分方程式常以其階數來分類)The most general form of first-order ODEs

8、is . (一階常微分方程式的通式)Note that must be present, but x and/or y need not occur explicitly. (方程式中，的成分必須存在，但x或y的成分未必存在)If a first-order ODE can be expressed as, (能表示成此種形式者叫做外顯式，否則叫做內隱式)It is called the explicit form (外顯式), in contrast with the implicit form (內隱式).A solution (解) of on an interval I is a fu

9、nction that satisfies the equation for all x in I. (能讓方程式成立的函數稱之為解) That is, for all x in I.A solution contained an arbitrary constant is called the general solution (通解) of the differential equation. (包含一個任意常數的一階微分方程式的解，稱之為通解)For example, is the general solution of .Each choice of the constant in t

10、he general solution yields a particular solution (特解).For example, is a particular solution of .A solution is explicit (外顯式的解) if it is given as a function of independent variables. For example, is an explicit solution of . Note that this general solution is explicit, with y isolated on one side of

11、an equation and a function of x on the other. (可以單獨把y孤立出來的解稱之為外顯式的解)By contrast, consider . The general solution is , which is implicit (內隱式的解). In this example we are unable to solve the differential equation explicitly for y as a function of x while isolating y on one side. (不能單獨把y孤立出來的解稱之為內隱式的解)T

12、he graph of a particular solution of a first-order differential equation is called an integral curve (積分曲線) or a solution curve (解的曲線) of the equation.Example 1 Verification of Solution (解的驗證)Example 2 Solution Curves (解的曲線)Example 3 Exponential Growth or Decay (指數型成長或衰減)If we specify that a particu

13、lar solution is a solution passing through a particular point (,), then we have to find that particular integral curve passing through this point. This is called an initial value problem (初始值問題).Thus, a first-order initial value problem has the form; ,in which and are given numbers. The condition is

14、 called an initial condition (初始條件).Example 4 Initial Value Problem Homework for sec.1.1 #7, 91.2 Direction Field (方向場) - Geometric Meaning of Consider the general first-order differential equation of the form.Suppose we can solve for as.Then a drawing of the plane, with short line segments of slope

15、 drawn at selected point (x,y), is called a direction field (方向場) or slope field (斜率場) of the differential equation.A direction field can be used to find approximate solutions, but with limited accuracy. (方向場可用於近似解之求取，儘管準確度較為有限)A famous ODE for which we do need direction field is.The direction field

16、 in Fig. 8 shows lineal elements generated by the computer. We have also added the isoclines (等斜率線) for , , , 1 as well as three typical solution curves, one that is a circle and two spirals approaching it from inside and outside. Homework for sec.1.2 #1, 5, 131.3 Separable ODEs (可分離型微分方程式) Many pra

17、ctically useful ODEs can be reduced to the separable form or .The variables are separated: x appears only on the right and y only on the left. And we can integrate the differential equation along any solution curve to get.In case f(x) and g(y) are continuous functions, we can obtain a general soluti

18、on from the above integrals.Example 1 A separable ODE (可分離型微分方程式)Example 2 Radiocarbon Dating (放射性碳定年)Example 3 Mixing Problem (混合問題)Certain non-separable ODEs can be made separable by transformations that introduce for y a new unknown function. (某些非可分離型微分方程式，可透過新函數的引進，將之轉換成可分離型)For a homogeneous OD

19、E, say,we can set to get and . Substituting them into the homogeneous ODE, we have ,which is a separable differential equation.Example 6 Reduction to Separable Form (轉換為可分離型) Homework for sec.1.3 #5, 9, 11, 13, 331.4 Exact ODEs (正合型微分方程式) If a function has continuous partial derivatives, its total d

20、ifferential (全微分) is.It follows that along any contour lines (等高線) of constant, we have.A first-order ODE written as or .is called exact (正合的) if there exists a function such that and .And we have .By integration, we immediate obtain the general solution of as.This is called an implicit solution (內隱

21、解), in contrast with an explicit solution (外顯解) of the form .Let and have continuous first partial derivatives. Then we have,.By the assumption of continuity, the two second partial derivatives are equal. Thus we have the following necessary and sufficient condition (充分必要條件). (Test for exactness)If

22、is exact, that is and , then the function can be found in the following systematic way:1) find ;2) find ;3) compare the results of 1) and 2) to determine ;4) the general solution is const.Example 1 An Exact ODE (正合型微分方程式之求解)Solve .An integrating factor (積分因子) is such a function that a multiplication

23、 of this factor with the differential equation will result in an equation that can be integrated to obtain the general solution.For the inexact (非正合) differential equation , the general integrating factor is provided is exact or .In case , we have In case , we have Homework for sec.1.4 #1, 5, 13, 17

24、, 21, 24(D)1.5 Linear ODEs (線性微分方程式) A first-order ODE is said to be linear (線性的) if it can be written as.(1)In engineering, is frequently called the input (輸入), and is called the output (輸出) or the response (反應;響應) to the input.For the linear differential equation , the general integrating factor i

25、s . Thus multiply eq.(1) on both sides by to get. Integrate this equation with respect to x to get Example 1 A Linear ODE (線性微分方程式之求解)Solve the linear ODE .Example 2 Initial Value Problem (初始值問題)Solve the initial value problem , .Numerous applications can be modeled by ODEs that are nonlinear but ca

26、n be transformed to linear ODEs. One of the most useful ones of these is the Bernoulli equation, a is any real number.(2)In case or , equation (2) is linear. Otherwise, it is nonlinear.To solve it, let a new function .Differentiate it on both sides to get , which is linear.Example 4 Logistic Equatio

27、n (數理邏輯方程式)Solve the Bernoulli equation, known as the logistic equation .(3)The logistic equation (3) plays an important role in population dynamics (人口動態學), a field that models the evolution of populations of plants, animals, or humans over time t. In case , the solution gives exponential growth as

28、 for a small population in a large country (the United States in early times!). The term in equation (3) is a braking term (抑制項) that prevents the population from growing without bound. Homework for sec.1.5 #3, 7, 11, 15, 18, 22, 271.6 Orthogonal Trajectories (正交軌跡)An important type of problem in ph

29、ysics or geometry is to find a family of curves that intersect a given family of curves at right angles. These new curves are called orthogonal trajectories (正交軌跡) of the given curves (and conversely). Orthogonal (正交) is another word for perpendicular (垂直).In many cases, orthogonal trajectories can be found by using ODEs as follows: Let be a given family of curves in the xy-plane, where each curve is specified by some value of c. This is ca