Chapter 1 First-Order Differential Equations

1. Chapter 1 First-Order Differential Equations A differential equation defines a relationship between an unknown function and one or more of its derivatives • Applicable to: • Chemistry • Physics • Engineering • Medicine • Biology Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

2. Chapter 1 First-Order Differential Equations 微分方程式的分類 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) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

3. Chapter 1 First-Order Differential Equations 常微分方程式 An ordinary differential equation is one with a single independent variable. The order (階) of an equation: The order of the highest derivative appearing in the equation First order differential equation with y as the dependent variable and x as the independent variable would be: Second order differential equation would have the form: Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

4. Chapter 1 First-Order Differential Equations • Linear – 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 linear Example of an Ordinary Differential Equation If f(x) = 0, The ODE is Homogeneous If f(x)  0, The ODE is Non-homogeneous 最高階導數的次數(degree)稱為此微分方程式之次數 Example : Degree = 1 Degree = 2 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

5. Chapter 1 First-Order Differential Equations 初值問題與邊界問題 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

6. ( ) ¢ = y cos x ( ) = + y sin x c general solution ( ) If c is defined ( particular solution ) Chapter 1 First-Order Differential Equations 通解與特殊解 • General Solution – • all solutions to the differential equation can be represented in this form • for all constants • Particular Solution – • contains no arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

7. 2 ¢ ¢ - + = y x y y 0 with general solution 2 = - y cx c 2 x = but parabola y 4 is also a solution Chapter 1 First-Order Differential Equations 奇異解(Singular Solutions) A differential equation may sometimes have an additional solution that cannot be obtained from the general solution. 顯式解(explicit solution) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

8. Chapter 1 First-Order Differential Equations 可分離微分方程式(Separable Differential Equations) Special form 分離變數(Separable Variables) Example: Boyle’s gas law Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

9. Chapter 1 First-Order Differential Equations Example 1 : Solve the differential equation 原式 : 兩邊作積分 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

10. Chapter 1 First-Order Differential Equations 試求 : 之通解 (83清大化工) 先利用座標平移來消去常數 2 與 –6 令 令 隱式解(implicit solution) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

11. Chapter 1 First-Order Differential Equations Reduction to Separable Form 1. Differential equations of the form --- 有時稱為齊次(homogeneous)方程式 其中g為y/x的任意函數,例如: , 欲求其解可令 and Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

12. Chapter 1 First-Order Differential Equations Reduction to Separable Form 2. Transformations Example : 代入並簡化 令 乘以2並分離變數 積分可得 隱式解(implicit solution) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

13. Chapter 1 First-Order Differential Equations Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

14. Chapter 1 First-Order Differential Equations First-Order Ordinary Differential Equations For φ(x,y)=C If We call Exact (正合) Differential Equation Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

15. Chapter 1 First-Order Differential Equations 試求 : 之解 (84中央光電) Exact Differential Equation Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

16. Chapter 1 First-Order Differential Equations Inexact Differential Equation If Exact Differential Equation But if Inexact Differential Equation An integrating factor Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

17. Chapter 1 First-Order Differential Equations 解 Inexact Differential Equation A function of x alone However, we see that Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

18. Chapter 1 First-Order Differential Equations First-Order Ordinary Differential Equations 亦可應用 全微分觀念 求通解 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

19. Chapter 1 First-Order Differential Equations First-Order Ordinary Differential Equations 亦可應用 全微分觀念 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

20. Chapter 1 First-Order Differential Equations 求通解 求通解 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

21. Chapter 1 First-Order Differential Equations Linear First-Order Ordinary Differential Equations An integrating factor We must require Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

22. Chapter 1 First-Order Differential Equations 試解線性微分方程式 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

23. Chapter 1 First-Order Differential Equations RL Circuit For special case : V(t) =V0 If the initial condition : I(0) = 0  C = -V0/R Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

24. Chapter 1 First-Order Differential Equations 解 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

25. Chapter 1 First-Order Differential Equations 解 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

26. Chapter 1 First-Order Differential Equations First-Order Ordinary Differential Equations (Nonlinear Linear) Nonlinear (Bernoulli equation) Linear Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

27. Chapter 1 First-Order Differential Equations 解 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

28. Chapter 1 First-Order Differential Equations 試解稱為Verhuls方程式的特殊柏努力方程式 (A,B為正值常數) (y = 0 也是一解) (稱為人口成長的logistic law) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

29. Chapter 1 First-Order Differential Equations (稱為人口成長的logistic law) 指數成長模型 (Malthus’s law) 當B = 0時 為一 “抑止項” (braking term) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

30. Chapter 1 First-Order Differential Equations q(x)  輸入(input), 例如 : 力 y(x)  輸出(output)或是響應(response), 例如 : 位移,電流….等 與初值相關 總輸出 = 對應於輸入的輸出 + 對應於初始數據的輸出 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

31. Chapter 1 First-Order Differential Equations Clairaut Differential Equations 通解 奇解(singular solution) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

32. 2 ¢ ¢ - + = y x y y 0 with general solution 2 = - y cx c 2 x = but parabola y 4 is also a solution Chapter 1 First-Order Differential Equations Singular Solution Envelope curve (包絡線) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

33. Chapter 1 First-Order Differential Equations 求通解與奇解 其中a, b 均為常數 通解 奇解(singular solution) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

34. Chapter 1 First-Order Differential Equations 一般情形下,若已知有包絡線存在,則對於曲線F(x,y,c)=0求出其包絡線 的方法是求出以下之聯立解 求通解與奇解 通解 奇解(singular solution) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

35. Chapter 8 Differential Equation First-Order ODE (Second-OrderFirst-Order) 1. F(x, y’, y’’) = 0, 意即不含因變數y 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x 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 取u(x) = y’/y  y’ = yu, y’’ = y’u + yu’ ( y’u + yu’) + pyu + qy = 0  u’ +[y’/y + p]u + q = 0 一階非線性常微分方程式 Riccati equation Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

36. Chapter 8 Differential Equation First-Order ODE --- Riccati equation setting v(x) is assumed to be a solution of the Riccati equation Linear First-Order Ordinary Differential Equations Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

37. Chapter 8 Differential Equation First-Order ODE --- Picard 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