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 歐拉-柯西 微分方程 二階線性齊次 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
a general solution will be of the form
, a linear combination of two solutions involving two arbitrary constants c1 and c2
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 y2 not 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 分析
自然數 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
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

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

51. Euler-Cauchy Case 3 :Example

52. Existence and Uniqueness Theory
If p(x) and q(x) are continuous function on some open interval  and x0 is in , then the initial value problem consisting of (1) and (3) has a unique solution y(x) on the interval .

53. Wronskian
A set of n functions y1(x), y2(x), …, yn(x), is said to be linearly dependent over an interval I if there exist n constants c1, c2, …, cn, not all zero, such that
Otherwise the set of functions is said to be linearly independent

54. Wronskian
A set of n functions y1(x), y2(x), …, yn(x), is linearly independent over an interval I if and only if the determinant (Wronski determinant, or Wronskian)

55. Wronskian
Example 3-8:
Sol:
 cosx, sinx are linearly independent

56. Linear Dependence and Independence of Solution
Suppose that (1) has continuous coefficients p(x) and q(x) on an open interval . Then two solutions y1 and y2 of (1) on  are linear dependent on  if and only if their Wronskian W is zero at some x0 in .
Furthermore, if W=0 for x= x0, then W=0 on ; hence if there is an x1in  at which W is not zero, then y1 ,y2 are liner independent on  .

57. Illustration of Theorem 2
Example 1
Example 2

58. A General Solution of (1) includes All Solutions
Theorem 3 (Existence of a general solution)
If p(x) and q(x) are continuous on an open interval , then (1) has a general solution on .
Theorem 4 (General solution)
Suppose that (1) has continuous coefficients p(x) and q(x) on some open interval . Then every solution y=Y(x) of (1) is of the form
where y1 , y2 form a basis of solutions of (1) on  and c1, c2 are suitable constants. Hence (1) does not have singular solutions (I.e., solutions not obtainable from a general solution)

59. Nonhomogeneous Equations
Theorem
(a) The difference of two solutions of (1) on some open interval  is a solution of (2) on 
(b) The sum of a solution of (1) and a solution of (2) on  is a solution of (1) on 

60. A general solution of the nonhomogeneous equation (1) on some open interval  is a solution of the form
where yh(x)=c1y1(x)+c2y2(x) is a general solution of the homogeneous equation (2) on  and yp(x) is any solution of (1) on  containing no arbitrary constants.
A particular solution of (1) on  is a solution obtain from (3) by assigning specific values to the arbitrary constants c1 and c2 in yh(x).

61. Practical Conclusion
To solve the nonohomegeneous equation (1) or an initial value problem for (1) , we have to solve the homogeneous equation (2) and find any particular solution yp of (1)

62. Initial value problem for a nonhomogeneous equation
Example

63. Solution by Undetermined Coefficients
Method of Undetermined Coefficients
General Solution: y = yh + yp
yh : Homogeneous Solution
yp : Particular Solution

64. Solution by Undetermined Coefficients

65. Rules for the Method of Undetermined Coefficients
Basic Rule
Modification Rule
Sum Rule

66. Solution by Undetermined Coefficients
Example 3-9:
Sol:

67. Example for Modification Rule
Example 1: in the case of a simple root
Example 2: in the case of a double root
Example 3: sum rule.

68. Second-Order Non-homogeneous Linear Equations
Method of Variation of Parameters
Particular Solution:
y1, y2 : Homogeneous Solutions
W : Wronskian of y1 and y2

69. Second-Order Non-homogeneous Linear Equations
Example 3-10:
Sol:

70. Higher Order Linear Differential Equations
Higher Order Homogeneous Linear ODE
If y1, y2, …, yn are the solutions of
y = c1y1+ c2y2 +… + cnyn will be the general solution

71. Higher Order Linear Differential Equations
Higher Order Nonhomogeneous Linear ODE
General Solution: y = yh + yp
yh : Homogeneous Solution
yp : Particular Solution