Chap 3 Linear Differential Equations

1. Chap 3 Linear Differential Equations 王 俊 鑫(Chun-Hsin Wang) 中華大學 資訊工程系 Fall 2002

2. Outline • Second-Order Homogeneous Linear Equations • Second-Order Homogeneous Equations with Constant Coefficients • Modeling: Mass-Spring Systems, Electric Circuits • Euler-Cauchy Equation • Wronskian • Second-Order Nonhomogeneous Linear Equations • Higher Order Linear Differential Equations

3. Outline 常係數 二階線性齊次 常微分方程 歐拉-柯西 微分方程 二階線性齊次 常微分方程 二階線性非齊次 常微分方程 高階線性 常微分方程 二階線性 常微分方程 線性 常微分方程 二階 常微分方程

4. Second-Order ODE • General Form for Second-Order Linear ODE • Implicit Form • Explicit Form

5. Second-Order Homogeneous Linear Equations • Second-Order Homogeneous Linear ODE • p(x), q(x): coefficient functions • Example

6. Examples of Nonlinear differential equations

7. A linear combination of Solutions for homogeneous linear equation • Example:

8. Second-Order Homogeneous Linear Equations • Linear Principle (Superposition Principle) • y is called the linear combination of y1 and y2 If y1 and y2 are the solutions of y = c1y1+ c2y2 is also a solution (c1, c2 arbitrary constants)

9. Second-Order Homogeneous Linear Equations Proof: Note

10. Does the Linearity Principle hold for nonhomogeneous linear or nonlinear equations ? • Example: A nonhomogeneous linear differential equation • Example: A nonlinear differential equation

11. Initial Value Problem for Second-Order homogeneous linear equations • For second-order homogeneous linear equations, a general solution will be of the form , a linear combination of two solutions involving two arbitrary constants c1 andc2 • An initial value problem consists two initial conditions.

12. Initial Value Problem • Example: • Observation: • Our solution would not have been general enough to satisfy the two initial conditions and solve the problem.

13. A General Solution of an Homogeneous Linear Equation • Definition: A general solution of an equation on an open interval I is a solution with y1 and y2not proportional solutions of the equation on I and c1 ,c2 arbitrary constants. • The y1 and y2 are then called a basis (or fundamental system) of the equation on I • A particular solution of the equation is obtained if we assign specific values to c1 ,c2

14. Linear Independent • Two functions y1(x) and y2(x) are linear independent on an interval I where they are defined if • Example

15. How to obtain a Bass if One Solution is Known ? • Method of Reduction Order • Given y1 • Find y2

16. Second-Order Homogeneous Linear Equations Proof:

17. Second-Order Homogeneous Linear Equations Proof:

18. Second-Order Homogeneous Linear Equations • Example 3-1: Sol:

19. Second-Order Homogeneous Linear Equations • Exercise 3-1: Basic Verification and Find Particular Solution Basis Initial Condition Basis Initial Condition Basis Initial Condition

20. Exercise: Reduce of order if a solution is known.

21. Second-Order Homogeneous Equations with Constant Coefficients • General Form of Second-Order Homogeneous Equations with Constant Coefficients whose coefficients a and b are constant.

22. Second-Order Homogeneous Equations with Constant Coefficients Sol: Characteristic Equation

23. Second-Order Homogeneous Equations with Constant Coefficients • Case 1: 兩相異實根 • Case 2: 重根 • Case 3: 共軛虛根

24. Second-Order Homogeneous Equations with Constant Coefficients • Example 3-2: Sol: Step 1: Find General Solution

25. Second-Order Homogeneous Equations with Constant Coefficients Step 2: Find Particular Solution

26. Second-Order Homogeneous Equations with Constant Coefficients Step 3: Plot Particular Solution MATLAB Code x=[0:0.01:2]; y=exp(x)+3*exp(-2*x); plot(x,y)

27. Case 2 Real Double Root = -a/2

28. Second-Order Homogeneous Equations with Constant Coefficients • Example 3-3: Sol: Step 1: Find General Solution

29. Second-Order Homogeneous Equations with Constant Coefficients Step 2: Find Particular Solution

30. Second-Order Homogeneous Equations with Constant Coefficients Step 3: Plot Particular Solution MATLAB Code x=[0:0.01:2]; y=(3-5*x).*exp(2*x); plot(x,y)

31. Euler Formula • Euler Formula Proof: Maclaurin Series

32. Euler Formula Proof:

33. Euler Formula 幾何 虛數 自然數 分析 負數

34. Complex Exponential Function

35. Case 3

36. Second-Order Homogeneous Equations with Constant Coefficients • Example 3-4: Sol: Step 1: Find General Solution

37. Second-Order Homogeneous Equations with Constant Coefficients Step 2: Find Particular Solution

38. Second-Order Homogeneous Equations with Constant Coefficients Step 3: Plot Particular Solution MATLAB Code x=[0:0.1:30]; y=exp(-0.1*x).*sin(2*x); plot(x,y)

39. Second-Order Homogeneous Equations with Constant Coefficients • Exercise 3-2: Find General Solution 兩相異實根 重根 共軛虛根

40. Modeling: Mass-Spring Systems

41. Modeling: Electric Circuits Capacitor (farads) Resistor (ohms) Inductor (heries)

42. Modeling

43. Modeling • Overdamping

44. Modeling • Critical Damping

45. Modeling • Underdamping

46. Euler-Cauchy Equation • Euler-Cauchy Equation The Auxiliary Equation

47. Euler-Cauchy Equation • Case 1: Distinct Real Roots m1, m2 • Example 3-5:

48. Euler-Cauchy Equation • Case 2: Double Roots m=(1-a)/2

49. Euler-Cauchy Case 2 :Example • Example

50. Euler-Cauchy Equation • Case 3: Complex Roots m = a ± bi