Chap 1 First-Order Differential Equations

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

2. Outline • Basic Concepts • Separable Differential Equations • substitution Methods • Exact Differential Equations • Integrating Factors • Linear Differential Equations • Bernoulli Equations

3. Basic Concepts • Differentiation

4. Basic Concepts • Differentiation

5. Basic Concepts • Integration

6. Basic Concepts • Integration

7. Basic Concepts • Integration

8. Basic Concepts • ODE vs. PDE • Dependent Variables vs. Independent Variables • Order • Linear vs. Nonlinear • Solutions

9. Basic Concepts • Ordinary Differential Equations • An unknown function (dependent variable) y of one independent variable x

10. Basic Concepts • Partial Differential Equations • An unknown function (dependent variable) z of two or more independent variables (e.g. x and y)

11. Basic Concepts • The order of a differential equation is the order of the highest derivative that appears in the equation. Order 2 Order 1 Order 2

12. Basic Concept • The first-order differential equation contain only y’ and may containyand given function of x. • A solution of a given first-order differential equation (*) on some open interval a<x<b is a function y=h(x) that has a derivative y’=h(x) and satisfies (*) for all x in that interval. (*) or

13. Basic Concept • Example : Verify the solution

14. Basic Concepts • Explicit Solution • Implicit Solution

15. Basic Concept • General solution vs. Particular solution • General solution • arbitrary constant c • Particular solution • choose a specific c

16. Basic Concept • Singular solutions • Def : A differential equation may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution. • Example The general solution : y=cx-c2 A singular solution : y=x2/4

17. Basic Concepts • General Solution • Particular Solution for y(0)=2 (initial condition)

18. Basic Concept • Def: A differential equation together with an initial condition is called an initial value problem

19. Separable Differential Equations • Def: A first-order differential equation of the form is called a separable differential equation

20. Separable Differential Equations • Example : Sol:

21. Separable Differential Equations • Example : Sol:

22. Separable Differential Equations • Example : Sol:

23. Separable Differential Equations • Example : Sol:

24. Separable Differential Equations • Substitution Method: A differential equation of the form can be transformed into a separable differential equation

25. Separable Differential Equations • Substitution Method:

26. Separable Differential Equations • Example : Sol:

27. Separable Differential Equations • Exercise 1

28. Exact Differential Equations • Def: A first-order differential equation of the form is said to be exact if

29. Exact Differential Equations • Proof:

30. Exact Differential Equations • Example : Sol:

31. Exact Differential Equations Sol:

32. Exact Differential Equations Sol:

33. Exact Differential Equations • Example

34. Non-Exactness • Example :

35. Integrating Factor • Def: A first-order differential equation of the form is not exact, but it will be exact if multiplied by F(x, y) then F(x,y) is called an integrating factor of this equation

36. Exact Differential Equations • How to find integrating factor • Golden Rule

37. Exact Differential Equations • Example : Sol:

38. Exact Differential Equations Sol:

39. Exact Differential Equations • Example :

40. Exact Differential Equations • Exercise 2

41. Linear Differential Equations • Def: A first-order differential equation is said to be linear if it can be written • If r(x) = 0, this equation is said to be homogeneous

42. Linear Differential Equations • How to solve first-order linear homogeneous ODE ? Sol:

43. Linear Differential Equations • Example : Sol:

44. Linear Differential Equations • How to solve first-order linear nonhomogeneous ODE ? Sol:

45. Linear Differential Equations Sol:

46. Linear Differential Equations • Example : Sol:

47. Linear Differential Equations • Example :

48. Bernoulli, Jocob Bernoulli, Jocob 1654-1705

49. Linear Differential Equations • Def: Bernoulli equations • If a = 0, Bernoulli Eq. => First Order Linear Eq. • If a <> 0, let u = y1-a

50. Linear Differential Equations • Example : Sol: