微分方程式的分类.ppt

《微分方程式的分类.ppt》由会员分享，可在线阅读，更多相关《微分方程式的分类.ppt（37页珍藏版）》请在得力文库 - 分享文档赚钱的网站上搜索。

1、Chapter 1 First-Order Differential EquationsA differential equation defines a relationship between an unknown function and one or more of its derivativesApplicable to:ChemistryPhysicsEngineeringMedicineBiologyY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung微分

2、方程式的分類Ordinary Differential Equation:(ODE)y 只與一個變數 x 有關Partial Differential Equation:(PDE)u 與兩個變數 x,y 有關Total Differential Equation:(TDE)Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungFirst order differential equat

3、ion with y as the dependent variable and x as the independent variable would be:Second order differential equation would have the form:常微分方程式常微分方程式An ordinary differential equation is one with a single independent variable.The order(階階)of an equation:The order of the highest derivative appearing in

4、the equationChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungLinear if the nth-order differential equation can be written:an(t)y(n)+an-1(t)y(n-1)+.+a1y+a0(t)y=h(t)Nonlinear not linearExample of an Ordinary Differenti

5、al EquationIf f(x)=0,The ODE is Homogeneous If f(x)0,The ODE is Non-homogeneous最高階導數的次數最高階導數的次數(degree)稱為此微分方程式之次數稱為此微分方程式之次數Example:Degree=1Degree=2Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Or

6、der Differential Equations初值問題與邊界問題初值問題與邊界問題Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungGeneral Solution all solutions to the differential equation can be represented in this form for all constantsParticular Solution contains no arbitrary constants()()()(

7、sincossolutionparticulardefinediscIfsolutiongeneralcxyxy+=Chapter 1 First-Order Differential Equations通解與特殊解通解與特殊解Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung奇異解奇異解(Singular Solutions)A differential equation may sometimes have an additional solution that

8、cannot be obtained from the general solution.solutionaalsoisxyparabolabutccxysolutiongeneralwithyyxy40222=-=+-Chapter 1 First-Order Differential Equations顯式解顯式解(explicit solution)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung可分離微分方程式可分離微分方程式(Separable Diffe

9、rential Equations)Special form分離變數分離變數(Separable Variables)Example:Boyles gas lawChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungExample 1:Solve the differential equationChapter 1 First-Order Differential Equations原

10、式原式:兩邊作積分兩邊作積分Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung試求試求:之通解之通解 (83清大化工清大化工)先利用座標平移來消去常數先利用座標平移來消去常數 2 與與 6 令令Chapter 1 First-Order Differential Equations隱式解隱式解(implicit solution)Y.M.Hu,Assistant Professor,Department of Applied Physics,National Univ

11、ersity of KaohsiungChapter 1 First-Order Differential EquationsReduction to Separable Form1.Differential equations of the form-有時稱為齊次有時稱為齊次(homogeneous)方程式方程式其中其中g為為y/x的任意函數的任意函數,例如例如:,欲求其解可令欲求其解可令 andY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1

12、First-Order Differential EquationsReduction to Separable Form2.Transformations Example:令令代入並簡化代入並簡化乘以乘以2並分離變數並分離變數積分可得積分可得隱式解隱式解(implicit solution)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Pro

13、fessor,Department of Applied Physics,National University of KaohsiungFirst-Order Ordinary Differential EquationsFor(x,y)=CIfWe call Exact(正合正合)Differential EquationChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung試求試

14、求:之解之解 (84 (84中央光電中央光電)Exact Differential EquationChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungInexact Differential EquationExact Differential EquationIfBut ifInexact Differential EquationAn integrating factorCha

15、pter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung解Inexact Differential EquationHowever,we see thatA function of x aloneChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics

16、,National University of KaohsiungFirst-Order Ordinary Differential Equations亦可應用亦可應用全微分觀念全微分觀念求通解Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungFirst-Order Ordinary Differential Equations亦可應用亦可應用全微分觀念全微分觀念Chapter 1

17、 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung求通解求通解Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungLinear First-Order Ordinary Different

18、ial EquationsAn integrating factorWe must requireChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Order Differential Equations試解線性微分方程式Y.M.Hu,Assistant Professor,Department of Applied Physics,National

19、 University of KaohsiungRL CircuitFor special case:V(t)=V0If the initial condition:I(0)=0 C=-V0/RChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung解Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professo

20、r,Department of Applied Physics,National University of Kaohsiung解Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungFirst-Order Ordinary Differential Equations(Nonlinear Linear)NonlinearLinearChapter 1 First-Order Diff

21、erential Equations(Bernoulli equation)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung解Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung試解稱為Verhuls方程式的特殊柏努力方程式 (A,B為正值常數)Chapt

22、er 1 First-Order Differential Equations(y=0 也是一解也是一解)(稱為人口成長的稱為人口成長的logistic law)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Order Differential Equations(稱為人口成長的稱為人口成長的logistic law)當當B=0時時指數成長模型指數成長模型(Malthuss law)為一為一“抑止項抑止項”(braking term

23、)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Order Differential Equationsq(x)輸入輸入(input),例如例如:力力y(x)輸出輸出(output)或是響應或是響應(response),例如例如:位移位移,電流電流.等等與初值相關與初值相關總輸出總輸出=對應於輸入的輸出對應於輸入的輸出+對應於初始數據的輸出對應於初始數據的輸出Y.M.Hu,Assistant Professor,Department

24、 of Applied Physics,National University of KaohsiungClairaut Differential Equations通解通解奇解奇解(singular solution)Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiungsolutionaalsoisxyparabolabutccxysolutiongeneralwithyyxy40

25、222=-=+-Singular SolutionEnvelope curve (包絡線)Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung求通解與奇解其中a,b 均為常數通解通解奇解奇解(singular solution)Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Departme

26、nt of Applied Physics,National University of Kaohsiung一般情形下一般情形下,若已知有包絡線存在若已知有包絡線存在,則對於曲線則對於曲線F(x,y,c)=0求出其包絡線求出其包絡線的方法是求出以下之聯立解的方法是求出以下之聯立解求通解與奇解通解通解奇解奇解(singular solution)Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kao

27、hsiungChapter 8 Differential EquationFirst-Order ODE(Second-OrderFirst-Order)1.F(x,y,y)=0,意即不含因變數意即不含因變數y取取p(x)=y F(x,y,y)=F(x,p,p)2.F(y,y,y)=0,意即不含自變數意即不含自變數x取取p(x)=y F(y,y,y)=F(y,p,pp)3.y+p(x)y+q(x)y=0 二階線性齊次常微分方程式二階線性齊次常微分方程式取取u(x)=y/y y=yu,y=yu+yu(yu+yu)+pyu+qy=0 u+y/y+pu+q=0一階非線性常微分方程式一階非線性常微分方

28、程式 Riccati equationY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 8 Differential EquationFirst-Order ODE-Riccati equationsettingv(x)is assumed to be a solution of the Riccati equationLinear First-Order Ordinary Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 8 Differential EquationFirst-Order ODE-Picard 疊代近似法疊代近似法對於初始值問題 ,若 可積分則其解為If 利用Picard疊代近似法計算到Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung