1. Introduction to Differential Equations CHAPTER 1

2. Lecturer: Prof. Hsin-Lung Wu Ch1_2 Chapter Contents  1.1 Definitions and Terminology 1.1 Definitions and Terminology  1.2 Initial-Value Problems 1.2 Initial-Value Problems  1.3 Differential Equations as Mathematical Methods 1.3 Differential Equations as Mathematical Methods

3. Lecturer: Prof. Hsin-Lung Wu Ch1_3 1.1 Definitions and Terminology  Introduction: differential equations means that equations contain derivatives, eg: dy/dx = 0.2xy (1)  Ordinary DE: An equation contains only ordinary derivates of one or more dependent variables of a single independent variable. eg: dy/dx + 5y = e x, (dx/dt) + (dy/dt) = 2x + y (2) An equation contains the derivates of one or more dependent variables with respect to one or more independent variables (DE). Definition 1.1.1 Differential equation

4. Lecturer: Prof. Hsin-Lung Wu Ch1_4  Partial DE: An equation contains partial derivates of one or more dependent variables of tow or more independent variables. (3)  Notations: Leibniz notation dy/dx, d 2 y/ dx 2 prime notation y’, y”, ….. subscript notation u x, u y, u xx, u yy, u xy, ….  Order: highest order of derivatives second order first order

5. Lecturer: Prof. Hsin-Lung Wu Ch1_5  General form of n-th order ODE: (4)  Normal form of (4) (5) eg: normal form of 4xy’ + y = x, is y’ = (x – y)/4x  Linearity: An n-th order ODE is linear if F is linear in y, y’, y”, …, y (n). It means when (4) is linear, we have (6)

6. Lecturer: Prof. Hsin-Lung Wu Ch1_6  The following cases are for n = 1 and n = 2 and (7)  Two properties of a linear ODE: (1) y, y’, y”, … are of the first degree. (2) Coefficients a 0, a 1, …, are at most on x  Nonlinear examples:

7. Lecturer: Prof. Hsin-Lung Wu Ch1_7  That is, a solution of (4) is a function  possesses at least n derivatives and F(x,  (x),  ’(x), …,  (n) (x)) = 0 for all x in I, where I is the interval  is defined on. Any function , defined on an interval I, possessing at least n derivatives that are continuous on I, when replaced into an n-th order ODE, reduces the equation into an identity, it said to be a solution of the equation on the interval. Definition 1.1.2 Solution of an ODE

8. Lecturer: Prof. Hsin-Lung Wu Ch1_8 Example 1 Verification of a Solution Verify the indicated function is a solution of the given ODE on ( － ,  ) (a) dy/dx = xy 1/2 ; y = x 4 /16 (b) Solution: (a) Left-hand side: Right-hand side: then left = right (b) Left-hand side: Right-hand side: 0 then left = right

9. Lecturer: Prof. Hsin-Lung Wu Ch1_9  Note: y = 0 is also the solution of example 1, called trivial solution

10. Lecturer: Prof. Hsin-Lung Wu Ch1_10 Example 2 Function vs. Solution y = 1/x, is the solution of xy’ + y = 0, however, this function is not differentiable at x = 0. So, the interval of definition I is ( － , 0), (0,  ). Fig 1.1.1 Ex 2 illustrates the difference between the function y = 1/x and the solution y = 1/x

11. Lecturer: Prof. Hsin-Lung Wu Ch1_11  Explicit solution: dependent variable is expressed solely in terms of independent variable and constants. Eg: solution is y =  (x). G(x, y) = 0 is said to be an implicit solution of (4) on I, provided there exists at least one function y =  (x) satisfying the relationship as well as the DE on I. Definition 1.3 Implicit solution of an ODE

12. Lecturer: Prof. Hsin-Lung Wu Ch1_12 Example 3 Verification of an Implicit Solution x 2 + y 2 = 25 is an implicit solution of dy/dx = −x/y (8) on the interval -5 < x < 5. Since dx 2 /dx + dy 2 /dx = (d/dx)(25) then 2x + 2y(dy/dx) = 0 anddy/dx = -x/y solution curve is shown in Fig 1.1.2

13. Lecturer: Prof. Hsin-Lung Wu Ch1_13 Fig 1.1.2 An Implicit solution and two explicit solutions in Ex 3

14. Lecturer: Prof. Hsin-Lung Wu Ch1_14  Families of solutions: A solution containing an arbitrary constant represents a set G(x, y) = 0 of solutions is called a one-parameter family of solutions. A set G(x, y, c 1, c 2, …, c n ) = 0 of solutions is called a n-parameter family of solutions.  Particular solution: A solution free of arbitrary parameters. eg: y = cx – x cos x is a solution of xy’ – y = x 2 sin x on (- ,  ), y = x cos x is a particular solution according to c = 0. See Fig 1.1.3.

15. Lecturer: Prof. Hsin-Lung Wu Ch1_15 Fig 1.1.3 Some solution of xy’-y=x 2 sinx

16. Lecturer: Prof. Hsin-Lung Wu Ch1_16 Example 4 Using Different Symbols x = c 1 cos 4t and x = c 2 sin 4t are solutions of x  + 16x = 0 we can easily verify that x = c 1 cos 4t + c 2 sin 4t is also a solution.

17. Lecturer: Prof. Hsin-Lung Wu Ch1_17 Example 5 A piecewise-Defined Solution We can verify y = cx 4 is a solution of xy – 4y = 0 on (- ,  ). See Fig 1.1.4(a). The piecewise-defined function is a particular solution where we choose c = −1 for x < 0 and c = 1 for x  0. See Fig 1.1.4(b).

18. Lecturer: Prof. Hsin-Lung Wu Ch1_18 Fig 1.1.4 Some solution of xy’-4y=0 in Ex 5

19. Lecturer: Prof. Hsin-Lung Wu Ch1_19  Singular solution: A solution can not be obtained by particularly setting any parameters. y = (x 2 /4 + c) 2 is the family solution of dy/dx = xy 1/2, however, y = 0 is a solution of the above DE.  We cannot set any value of c to obtain the solution y = 0, so we call y = 0 is a singular solution.

20. Lecturer: Prof. Hsin-Lung Wu Ch1_20  System of DEs: two or more equations involving of two or more unknown functions of a single independent variable. dx/dt = f(t, x, y) dy/dt = g(t, x, y) (9)

21. Lecturer: Prof. Hsin-Lung Wu Ch1_21 1.2 Initial-value Problems  Introduction A solution y(x) of a DE satisfies an initial condition.  Example On some interval I containing x o, solve: subject to: (1) This is called an Initial-Value Problem (IVP). y(x o ) = y o, y(x o ) = y 1, are called initial conditions.

22. Lecturer: Prof. Hsin-Lung Wu Ch1_22  First and Second IVPs (2) and (3) are first and second order initial-value problems, respectively. See Fig 1.2.1 and 1.2.2.

23. Lecturer: Prof. Hsin-Lung Wu Ch1_23

24. Lecturer: Prof. Hsin-Lung Wu Ch1_24 Example 1 First-Order IVPs We know y = ce x is the solutions of y’ = y on (- ,  ). If y(0) = 3, then 3 = ce 0 = c. Thus y = 3e x is a solution of this initial-value problem y’ = y, y(0) = 3. If we want a solution pass through (1, -2), that is y(1) = -2, -2 = ce, or c = -2e -1. The function y = -2e x-1 is a solution of the initial-value problem y’ = y, y(1) = -2.

25. Lecturer: Prof. Hsin-Lung Wu Ch1_25 In Problem 6 of Exercise 2.2, we have the solution of y’ + 2xy 2 = 0 is y = 1/(x 2 + c). If we impose y(0) = -1, it gives c = -1. Consider the following distinctions. 1) As a function, the domain of y = 1/(x 2 - 1) is the set of all real numbers except x = -1 and 1. See Fig 1.2.4(a). 2) As a solution, the intervals of definition are (- , 1), (-1, 1), (1,  ) 3) As a initial-value problem, y(0) = -1, the interval of definition is (-1, 1). See Fig 1.2.4(b). Example 2 Interval / of Definition of a Solution

26. Lecturer: Prof. Hsin-Lung Wu Ch1_26 Fig 1.2.4 Graph of the function and solution of IVP in Ex 2

27. Lecturer: Prof. Hsin-Lung Wu Ch1_27 Example 3 Second-Order IVP In Example 4 of Sec 1.1, we saw x = c 1 cos 4t + c 2 sin 4t is a solution of x  + 16x = 0 Find a solution of the following IVP: x  + 16x = 0, x(  /2) = −2, x(  /2) = 1 (4) Solution: Substitute x(  /2) = − 2 into x = c 1 cos 4t + c 2 sin 4t, we find c 1 = −2. In the same manner, from x(  /2) = 1 we have c 2 = ¼.

28. Lecturer: Prof. Hsin-Lung Wu Ch1_28  Existence and Uniqueness: Does a solution of the IVP exist? If a solution exists, is it unique?

29. Lecturer: Prof. Hsin-Lung Wu Ch1_29 Example 4 An IVP Can Have Several Solution Since y = x 4 /16 and y = 0 satisfy the DE dy/dx = xy 1/2, and also initial-value y(0) = 0, this DE has at least two solutions, See Fig 1.2.5.

30. Lecturer: Prof. Hsin-Lung Wu Ch1_30 Let R be the region defined by a  x  b, c  y  d that contains the point (x 0, y 0 ) in its interior. If f(x, y) and  f/  y are continuous in R, then there exists some interval I 0 : (x 0 - h, x 0 + h), h > 0, contained in [a, b] and a unique function y(x) defined on I 0 that is a solution of the IVP (2). Theorem 1.2.1 Existence of a Unique Solution

31. Lecturer: Prof. Hsin-Lung Wu Ch1_31 Fig 1.2.6 Rectangular region R  The geometry of Theorem 1.2.1 shows in Fig 1.2.6.

32. Lecturer: Prof. Hsin-Lung Wu Ch1_32 Example 5 Example 3 Revisited For the DE: dy/dx = xy 1/2, inspection of the functions we find they are continuous in y > 0. From Theorem 1.2.1, we conclude that through any point (x 0, y 0 ), y 0 > 0, there is some interval centered at x 0 on which this DE has a unique solution.

33. Lecturer: Prof. Hsin-Lung Wu Ch1_33  Interval of Existence / Uniqueness Suppose y(x) is a solution of IVP (2), the following sets may not be the same:  the domain of y(x), the interval of definition of y(x) as a solution, the interval I 0 of existence and uniqueness.

34. Lecturer: Prof. Hsin-Lung Wu Ch1_34 1.3 DEs as Mathematical Models  Introduction Mathematical models are mathematical descriptions of something.  Level of resolution Make some reasonable assumptions about the system.  The steps of modeling process are as following.

35. Lecturer: Prof. Hsin-Lung Wu Ch1_35 Assumptions Mathematics formulation Check model Predictions with know facts Obtain solution Express assumptions in terms of differential equations If necessary, alter assumptions or increase resolution of the model Solve the DEs Display model predictions, e.g., graphically

36. Lecturer: Prof. Hsin-Lung Wu Ch1_36  Population Dynamics If P(t) denotes the total population at time t, then dP/dt  P or dP/dt = kP (1) where k is a constant of proportionality, and k > 0.  Radioactive Decay If A(t) denotes the substance remaining at time t, then dA/dt  A or dA/dt = kA (2) where k is a constant of proportionality, and k < 0.  A single DE can serve as a mathematical model for many different phenomena.

37. Lecturer: Prof. Hsin-Lung Wu Ch1_37  Newton’s Law of Cooling/Warming If T(t) denotes the temperature of a body at time t, T m the temperature of surrounding medium, then dT/dt  T - T m or dT/dt = k(T - T m ) (3) where k is a constant of proportionality.

38. Lecturer: Prof. Hsin-Lung Wu Ch1_38  Spread of a Disease If x(t) denotes the number of people who have got the disease and y(t) the number of people who have not yet, then dx/dt = kxy (4) where k is a constant of proportionality. From the above description, suppose a small community has a fixed population on n, If one inflected person is introduced into this community, we have x + y = n +1, and dx/dt = kx(n+1-x) (5)

39. Lecturer: Prof. Hsin-Lung Wu Ch1_39  Chemical Reactions Inspect the following equation CH 3 Cl + NaOH  CH 3 OH + NaCl Assume X is the amount of CH 3 OH,  and  are the amount of the first two chemicals, then the rate of formation is dx/dt = k(  - x)(  - x) (6)

40. Lecturer: Prof. Hsin-Lung Wu Ch1_40  Mixtures See Fig 1.3.1. If A(t) denotes the amount of salt in the tank at time t, then dA/dt = (input rate) – (output rate) = R in - R out (7) We have R in = 6 lb/min, R out = A(t)/100 (lb/min), then dA/dt = 6 – A/100 or dA/dt + A/100 = 6 (8)

41. Lecturer: Prof. Hsin-Lung Wu Ch1_41 Fig 1.3.1 Mixing tank

42. Lecturer: Prof. Hsin-Lung Wu Ch1_42  Draining a Tank Referring to Fig 1.3.2 and from Torricelli’s Law, if V(t) denotes the volume of water in the tank at time t, (9) From (9), since we have V(t) = A w h, then (10)

43. Lecturer: Prof. Hsin-Lung Wu Ch1_43 Fig 1.3.2 Water draining from a tank

44. Lecturer: Prof. Hsin-Lung Wu Ch1_44  Series Circuits Look at Fig 1.3.3. From Kirchhoff’s second law, we have (11) where q(t) is the charge and dq(t)/dt = i(t), which is the current.

45. Lecturer: Prof. Hsin-Lung Wu Ch1_45 Fig 1.3.3 Current i(t) and charge q(t) are measured in amperes (A) and coulumbs (C)

46. Lecturer: Prof. Hsin-Lung Wu Ch1_46  Falling Bodies Look at Fig1.22. From Newton’s law, we have (12) Initial value problem (13)

47. Lecturer: Prof. Hsin-Lung Wu Ch1_47 Fig 1.3.4 Position of rock measured from ground level

48. Lecturer: Prof. Hsin-Lung Wu Ch1_48  Falling Bodies and Air Resistance From Fig 1.3.5. We have the DE (14) and can be written as or (15)

49. Lecturer: Prof. Hsin-Lung Wu Ch1_49 Fig 1.3.5 Falling body of mass m

50. Lecturer: Prof. Hsin-Lung Wu Ch1_50  A Slipping Chain From Fig 1.3.6. We have (16)

51. Lecturer: Prof. Hsin-Lung Wu Ch1_51 Fig 1.3.6 Chain slipping from frictionless peg

52. Lecturer: Prof. Hsin-Lung Wu Ch1_52  Suspended Cables From Fig1.25. We have dy/dx = W/T 1 (17)

53. Lecturer: Prof. Hsin-Lung Wu Ch1_53 Fig 1.3.7 Cables suspended between vertical supports

54. Lecturer: Prof. Hsin-Lung Wu Ch1_54 Fig 1.3.8 Element of cable  Fig 1.3.8 explains the Element of cable.

55. Lecturer: Prof. Hsin-Lung Wu Ch1_55 Evaluation  One midterm 40%  Final 40%  Quiz & Homework 20%

56. Lecturer: Prof. Hsin-Lung Wu Ch1_56 Some information  Textbook: Advanced Engineering Mathematics 4 th -ed by Zill and Wright Teaching Website: Please link 數位學苑