1. 1 Part 1: Ordinary Differential Equations Ch1: First-Order Differential Equations Ch2: Second-Order Differential Equations Ch3: The Laplace Transform Ch4: Series Solutions

2. (a)Differential equation --- contains derivatives e.g., or or y : function of x, x : independent variable (b) Ordinary differential equation – involves only total derivatives Partial differential equation – involves partial derivatives What is 2

3. (c) The order of a differential equation e.g., ○ The solution of a differential equation is a function y(x) of independent variable x that may be defined on e.g., i), solution: y = sin2x for ii), solution: y = xlnx – x for x > 0 3

4. Ch. 1: First-Order Differential Equations 1.1. Preliminary Concepts ○ First-order differential equation: -- involves a first but no higher derivatives e.g., y ： function of x x ： independent variable : solution 4

5. 1.1.1. General and Particular Solutions ○ General solution: arbitrary constant Substitute into (A) Particular solutions: k = 1, ; k = 2, k =, 5

6. 1.1.2. Implicitly Defined Solutions ○ Explicit function: Implicit function: e.g.,, ○ (Explicit solution) (Implicit solution) 6

7. 1.1.3. Integral Curves -- Help to comprehend the behavior of solution ○ Example 1.1: General solution: 7

8. 1.1.4. Initial Value Problems ○ initial condition Graphically, the particular integral curve passes through point ( ) The objective is to obtain a unique solution ○ Example 1.4:, initial condition General solution:, 8

9. 1.1.5. Direction Field -- A set of line segments tangent to a curve -- Give a rough outline of the shape of the curve ○ Giving, instead of solving for y, solving for 9

10. 1.1.5. Direction Field -- A set of line segments tangent to a curve -- Give a rough outline of the shape of the curve ○ Giving, instead of solving for y, solving for 10

11. ○ Example 1.5: Slope:, General Solution: Figure 1.5. Direction field for y'= y² and integral curves through (0,1), (0,2), (0,3), - 1, (0, - 2), and (0, - 3). 11

12. 1.2. Separable Equations 12

13. ○ Example 1.7:, (the general solution) y = 0 is a solution, called a singular solution, it cannot be obtained from the general solution 13

14. 1.3. Linear Differential Equations -------- (A) (1)Find integrating factor: (2)Multiply (A) by (3) 14

15. ○ Example 1.14: i) Integrating factor ii) Multiply the equation by, iii) iv) Integrate 15

16. 1.4. Exact Differential Equations can be written as ------- (A) If, s.t. and (implicitly define the solution) : potential function 16

17. ○ Example 1.17: Let From 17

18. From Solution: 18 ＊ We can start with, then as well

19. ＊ Not every is exact e.g. From c(y) is not independent of x No potential function 19

20. Theorem 1.1: Exactness is exact iff (a) If is exact, then s.t. 20 (b) If, show s.t.

21. 21 Let : any point

22. ○ Example : is not exact 22 1.5. Integrating Factors If : not exact But : exact integrating factor

23. ○ How to find : exact, Try as, or 23 ○ Example 1.21:. The equation is not exact

24. Consider Let Let Try (B) Integrate 24

25. (A) Let be the potential function The implicit solution: The explicit solution: 25

26. ○ Example 1.22: Let Find integrating factor by 26

27. (i)Try 27 (ii) Likewise, let This cannot be solved for as a function of x

28. (iii) Try (B) Divide by 28

29. : independent Multiply (A) by Let Form 29

30. From Obtain the potential function The implicit solution: or 30

31. 1.6. Homogeneous, Bernoulli, and Riccati Eqs. 1.6.1 Homogeneous Equation: ----- (A) ＊ A homogeneous equation is always transformed into a separable one by letting ↑=1 (A) 31

32. ○ Example 1.25: Let (A) 32

33. 1.6.2. Bernoulli Equation linear separable can be transformed into linear by 33

34. ○ Example 1.27: Let (A) Multiply by (linear) Integrating factor: 34

35. 1.6.3. Riccati Equation Let S(x) be a solution and let The Riccati equation is transformed into linear ○ Example 1.28: By inspection, is a solution of (A) Let 35

36. (linear) Integrating factor: Integrate and Solution: 36

37. 1.8. Existence and Uniqueness for Solutions of Initial Value Problems Initial Value Problems: The problem may have no solution and may have multiple solutions ○ Example 1.30: The equation is separable and has solution The equation has no real solution. is not a solution because it does not satisfy the initial condition. 37

38. ○ Example 1.31: The equation has solution This problem has multiple solutions i, Trivial solution: ii, Define Consider All satisfy the initial condition 38

39. ○ Theorem 1.2 ： If f, : continuous in, then s.t. The initial value problem has a unique solution defined on The size of h depends on f and 39

40. ○ Example 1.31: 40 not continuous on (x, 0) ○ Example ： The problem : both continuous on the entire plane and hence on s.t. the problem has a unique solution in Solve the problem, which is valid for, we can take

41. ○ Theorem 1.3 ： : continuous on I The problem, has a unique solution defined Proof ： From the general solution of the linear equation and the initial condition, the solution of the initial value problem is 41 : continuous on I, the solution is defined

42. 42 Homework 1 Chapter 1 Sec.1.1: 1, 2, 7, 12 Sec.1.2: 1, 2, 11 Sec.1.3: 1 Sec.1.4: 1 Sec.1.5: 1

43. 43 1.5(1): Determine a test involving M and N to tell when has an integrating factor that is a function of y only. Ans: Let be an integrating factor such that is exact. Then,

44. 44 The test is then that must be independent of x.

45. 45 Homework 2 Chapter 1 Sec.1.6: 15, 16, 20, 21 Sec.1.8: 1, 3, 5