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 contain y and 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
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

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:

51. Linear Differential Equations
Exercise 3

52. Summary

53. Orthogonal Trajectories of Curves
Angle of intersection of two curves is defined to be the angle between the tangents of the curves at the point of intersection
How to use differential equations for finding curves that intersect given curves at right angles ?

54. How to find Orthogonal Trajectories
1st Step: find a differential equation
for a given cure
2nd Step: the differential equation of the orthogonal trajectories to be found
3rd step: solve the differential equation as above ( in 2nd step)

55. Orthogonal Trajectories of Curves
Example: given a curve y=cx2, where c is arbitrary. Find their orthogonal trajectories.
Sol:

56. Existance and Uniqueness of Solution
An initial value problem may have no solutions, precisely one solution, or more than one solution.
Example
No solutions
Precisely one solutions
More than one solutions

57. Existence and uniqueness theorems
Problem of existence
Under what conditions does an initial value problem have at least one solution ?
Existence theorem, see page 53
Problem of uniqueness
Under what conditions does that the problem have at most one solution ?
Uniqueness theorem, see page54