1. Differential Equations Also known as Engineering Analysis or ENGIANA

2. Introduction In science, engineering, economics, and in most areas having a quantitative component, we are interested in describing how systems evolve in time, that is, in describing a system’s dynamics.

3. Introduction In this time series plot of a generic state function u = u(t) for a system, one can monitor the state of a system (e.g. population, concentration, temperature, etc) as a function of time. However, such curves or formulas tell us how a system behaves in time, but they do not give us insight into why a system behaves in the way we observe.

4. Introduction Thus, we try to formulate explanatory models that underpin the understanding we seek. Often these models are dynamic equations that relate the state u(t) to its rates of change, as expressed by its derivatives u′(t), u′′(t),..., and so on.

5. Introduction Before we present the formal definition, let’s see some examples of differential equations:

6. Introduction The first equation models the angular deflections θ = θ(t) of a pendulum of length L.

7. Introduction The second equation models the charge q = q(t) on a capacitor in an electrical circuit containing a resistor and a capacitor, where the current is driven by a sinusoidal electromotive force sin ωt operating at frequency ω.

8. Introduction In the third equation, called the logistic equation, the state function p = p(t) represents the population of an animal species in a closed ecosystem; r is the population growth rate and K represents the capacity of the ecosystem to support the population.

9. Introduction The fourth equation represents a model of motion, where x = x(t) is the position of a mass acted upon by a force −αx.

10. Introduction Differential Equation –An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation (DE). Classification –By Type –By Order –By Linearity

11. Ordinary Differential Equations If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable, it is said to be an ordinary differential equation (ODE).

12. Partial Differential Equations An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a partial differential equation (PDE).

13. Notation on ODEs Leibniz notation Prime notation Newton’s Dot Notation

14. Notation on PDEs Leibniz Notation Subscript Notation

15. Order of a Differential Equation The order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. second order

16. Order of a Differential Equation The order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. first order

17. First Order ODEs First-order ordinary differential equations are occasionally written in the differential form M(x, y)dx + N(x, y)dy = 0. For example:

18. General and Normal Forms of ODEs In symbols, we can express an nth-order ordinary differential equation in one dependent variable by the general form where F is a real-valued function of n+2 variables x, y, y’, …, y (n).

19. General and Normal Forms of ODEs In symbols, we can express an nth-order ordinary differential equation in one dependent variable by the normal form where f is a real-valued continuous function.

20. General and Normal Forms of ODEs Example of ODE in general form: Example of ODE in normal form:

21. Linearity of ODEs An nth-order ordinary differential equation is said to be linear if F is linear in y, y’, …, y (n) :

22. Linearity of ODEs First-order ODE: Second-order ODE:

23. Linearity of ODEs The dependent variable y and all its derivatives y’, y’’, …, y (n), etc. are of the first degree (that is, the power of each said term involving y is 1). The coefficients a 0, a 1, …, a n of y, y’,…,y n depend at most on the independent variable x.

24. Linearity of ODEs A linear, first order ODE: A linear, second order ODE A linear, third order ODE

25. Nonlinear ODE A nonlinear ordinary differential equation is simply one that is not linear. Nonlinear functions of the dependent variable or its derivatives, such as sin( y) or e y’, cannot appear in a linear equation.

26. Nonlinear ODEs A nonlinear, first order ODE: A nonlinear, second order ODE A nonlinear, fourth order ODE Nonlinear term: coefficient depends on y Nonlinear term: nonlinear function of y Nonlinear term: power not 1

27. Nonlinear ODEs A nonlinear ODE Nonlinear term: power not 1

28. Solution of an ODE Any function , defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval.

29. Solution of an ODE In other words, a solution of an nth-order ordinary differential equation is a function  that possesses at least n derivatives and for which for all x in I.

30. The Interval of the Solution The Interval of the Solution is also called –Interval of definition –Interval of existence –Interval of validity –Domain of the solution

31. Solution Curve The graph of a solution  of an ordinary differential equation is called a solution curve. Since  is a differentiable function, it is continuous on its interval I of definition.

32. Families of Solutions The study of differential equations is similar to that of integral calculus. In some texts, a solution  is sometimes referred to as an integral of the equation and its graph is called an integral curve.

33. Families of Solutions When evaluating an anti-derivative or indefinite integral in calculus, we use a single constant c of integration. Analogously, when solving a first-order differential equation F(x, y, y’) = 0, we usually obtain a solution containing a single arbitrary constant or parameter c.

34. Families of Solutions A solution containing an arbitrary constant represents a set G(x, y, c) = 0 of solutions called a one-parameter family of solutions. When solving an nth-order differential equation F(x, y, y’, …, y (n) ) = 0, we seek an n-parameter family of solutions. –We integrate n times, hence we get n arbitrary constants.

35. Families of Solutions This means that a single differential equation can possess an infinite number of solutions corresponding to the unlimited number of choices for the parameter(s). A solution of a differential equation that is free of arbitrary parameters is called a particular solution.

36. Families of Solutions For example, the one-parameter family y = cx – xcosx is a solution of the linear first-order equation xy’ – y = x 2 sinx on the interval (- ,  ).

37. Families of Solutions Check: y = cx – xcosx y’ = c – ( cosx + x(-sinx) ) y’ = c – cosx + xsinx xy’ – y = x 2 sinx x(c – cosx + xsinx) – (cx – xcosx) = x 2 sinx cx – xcosx + x 2 sinx – cx + xcosx = x 2 sinx x 2 sinx = x 2 sinx

38. Families of Solutions Some solutions of xy’ – y = x 2 sinx Graph of y = cx – xcosx

39. Families of Solutions If every solution of an nth-order ODE F(x, y, y’, …, y (n) ) on an interval I can be obtained from an n-parameter family G(x, y, c 1, c 2, …, c n ) = 0 by appropriate choices of the parameters c i, i = 1, 2, …, n, we then say that the family is the general solution of the differential equation.

40. Singular Solution Sometimes a differential equation possesses a solution that is not a member of a family of solutions of the equation – that is, a solution that cannot be obtained by specializing any of the parameters in the family of solutions. Such an extra solution is called a singular solution.

41. Singular Solution For example, is a solution of the differential equation on the interval (- ,  ).

42. Singular Solution When c = 0, the resulting particular solution is But notice that the trivial solution y = 0 is a singular solution, since it is not a member of the family y = (1/2x 2 + c) 2 ; there is no way of assigning a value to the constant c to obtain y = 0.

43. Initial-Value Problems (IVPs) We are often interested in problems in which we seek a solution y(x) of a differential equation so that y(x) satisfies prescribed side conditions – that is, conditions imposed on the unknown y(x) or its derivatives.

44. Initial-Value Problems (IVPs) Solve: Subject to: where y 0, y 1, …, y n-1 are arbitrarily specified real constants.

45. Initial-Value Problems (IVPs) Such a problem is called an initial-value problem (IVP). The values of y(x) and its first n-1 derivatives at single point x 0 (i.e., y(x 0 ) = y 0, y’(x 0 ) = y 1, …, y (n-1) (x 0 ) = y n-1 ) are called initial conditions.

46. First-Order IVP Solve: Subject to:

47. Second-Order IVP Solve: Subject to: m = y 1

48. Example What function do you know from calculus is such that its first derivative is itself?

49. Example Answer: y = Ce x y = Ce x so that y’ = Ce x and hence, y = y’ If we impose the following initial condition y(0) = 3 then we have y = Ce x 3 = Ce 0 3 = C

50. Example Hence, y = 3e x is a solution of the IVP y’ = y y(0) = 3 If we impose the following initial condition y(1) = -2 then we have y = Ce x -2 = Ce 1 -2/e = C

51. Example Hence, y = (-2/e)e x or y = -2e x-1 is a solution of the IVP y’ = y y(1) = -2 The two solution curves are shown in dark blue and dark red in the next figure.

52. Example The solution curves for and

53. Existence of a Unique Solution Theorem Let R be a rectangular region in the xy- plane 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 on R, then there exist some interval I 0 : (x o - h, x o + h), h > 0, contained in [a, b], and a unique function y(x), defined on I 0, that is a solution of the initial-value problem.

54. Existence of a Unique Solution

55. Direction or Slope Fields Recall that a derivative dy/dx of a differentiable function y = y(x) gives slopes of tangent lines at points on its graph. The derivative itself is also a function. In other words,

56. Direction or Slope Fields The function f in the normal form is called the slope function or rate function. The value f(x,y) that the function f assigns to a point (x,y) represents the slope of a line; alternatively, we can envision it as a line segment called a lineal element.

57. Direction or Slope Fields For example, say we’re given the following function f and a point (2, 3) on the solution curve: At the point (2, 3), the slope of a lineal element is f(2,3) = 2(2)(3) = 1.2.

58. A lineal element at a pointA lineal element is tangent to the solution curve that passes through the point in consideration

59. Direction or Slope Fields If we systematically evaluate f over a rectangular grid of points in the xy plane and draw a line element at each point (x,y) of the grid with slope f(x,y), then the collection of all these lines is called a direction field or a slope field of the differential equation dy/dx = f(x,y).

60. Direction or Slope Fields Visually, the direction field suggests the appearance or shape of a family of solution curves of the differential equation, and consequently, it may be possible to see at a glance certain qualitative aspects of the solutions.

61. A single solution curve that passes through a direction field must follow the flow pattern of the field. It is tangent to a line element when it intersects a point in the grid. This figure shows a computer generated direction field of dy/dx = sin(x+y) over a region of the xy plane. Note how the solution curves shown in color follow the flow of the field.

62. Direction or Slope Fields Going back to it can be shown that is a one-parameter family of solutions.

63. Direction Field for dy/dx = 0.2xy Some solution curves in the family

64. Solving Differential Equations 1.Analytical Approach 2.Qualitative Approach 3.Numerical Approach

65. Solving Differential Equations 1.Analytical Approach –Exact solution using mathematical principles (calculus) –Main focus of this course (ENGIANA)

66. Solving Differential Equations 2.Qualitative Approach –Gleaning from the differential equation answers to questions such as Does the DE actually have solutions? If a solution of the DE exists and satisfies an initial condition, is it the only such solution? What are some of the properties of the unknown solutions? What can we say about the geometry of the solution curves?

67. Solving Differential Equations 3.Numerical Approach –Use of numerical methods and computer algorithms Can we somehow approximate the values of an unknown solution?