DE for ECE Intro ODE PDE, Var Sep, Homo.pptx

DE for ECE

1. Class Orientation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

2. Highlights from the Student Handbook TITLE II Chap. 8 Attendance of Students Article 1: “show proof of registration before being admitted to class” Article 2: “tardy when he/she arrives past 25% of the scheduled class period.” 45 mins late “Three instances of tardiness is equivalent to one absence.” “In cases where instructors are late for class, students should not leave the classroom until the first third fraction … has passed.” 1 hour Article 3: “Absences of students may be excused for: Section 1: Illness. Student must submit a medical certificate issues by the University Physician. If by other physician, have it authenticated by the University Physician. Section 2: Natural calamities Section 3: Official participation of approved co- or extra-curricular events. Student must submit an approval form to the intstructor . Note: walay labot ang mamatyan ug relative (not excused) UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

3. Highlights from the Student Handbook TITLE II Chap . 13 Examinations Article 2: student may be given late examinations within one week. . .for illness, accident or death of an immediate member of the family. Present a medical or death certificate to be validated by the University Physician. Article 3: In case a student incurs a conditional grade (3.25 to 3.5), instructor concerned shall give a removal examination (70% passing) within the week after the scheduled final examination UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

4. Highlights from the Student Handbook TITLE IV Chap. 5 Offenses and Penalties Article 1: Academic Offense: Cheating – any deceitful, fraudulent or dishonest act of a student which shows lack of integrity and a disposition to lie, betray and violate the truth UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

5. Differential Equations UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

6. Inspiration: We live in a changing world! position of earch changes with time velocity of a falling body changes with distance area of circle changes with radius Changing entities (time, position, temperature) are called variables . Rate of change of one variable with respect to another is derivative . UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

7. Mathematical model - a mathematical equation expressing the essential features of a physical system or process in order to approximate real-world problems Many physical laws and relations can be modelled mathematically in the form of differential equations . UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

8. Find all the derivatives of a. b. Find the partial derivatives and of A review on Differential Calculus (Math111) Differential Equation - an equation involving variables and their derivatives UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

9. Consider y = x 2 and its first and second derivatives, what is y'' + 2 y' + y ? This course tells us how to do the reverse process! That is, given the differential equation we will be able to get a solution of this equation which is UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

10. Examples of Differential Equations UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

11. When an equation involves one or more derivatives with respect to a particular variable, that variable is called an independent variable . A variable is called dependent if a derivative of that variable occurs. t - independent variable i - dependent variable Independent and Dependent Variable UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

12. Consider a function y = f ( x ). Ordinary Differential Equation (ODE) is a differential equation involving only one independent variable x , the function f ( x ) and one or more of the derivatives of f ( x ). Examples: Partial Differential Equation (PDE) involves partial derivatives of an unknown function of two or more variables. Example: where u = f ( x , y ) Ordinary and Partial Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

13. The focus of this course is on ODE's. Therefore, from now on by saying differential equation, we mean ODE. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

14. The order of a differential equation is the order of the highest derivative appearing in the equation. Example: Order of a differential equation second-order equation first-order eqution UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

15. The degree of a differential equation is the power (exponent) of its highest derivative appearing in the equation provided that 1. there are no fractional powers 2. the derivatives must not be involved in a fraction 3. the highest-order derivative must not be involved as a transcendental (non-polynomial) function Example: Degree of a differential equation third degree second degree UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

16. Exercise: Find the degree and order of the ff. DE 1st order, 1st degree 2nd order, 1st degree 2nd order, 3rd degree 2nd order, 1st degree 3rd order, Degree is not defined 3rd order, 1st degree UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

17. Solution of a Differential Equation Let y = f ( x ) be a function defined on an interval I . f ( x ) is a solution of an ODE involving x , f ( x ) and its derivatives if, replacing y by f ( x ), y' by f' ( x ), y'' by f'' ( x ), so on and so fort, the differential equation reduces to an identity in x . Remember! The solution of an ODE is a function . UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

18. Example: Verify that the function y = x 2 is a solution of the differential equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

19. Example: Verify that the function defined by is a solution of the differential equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

20. DOING THE REVERSE: Example: Using differential calculus, find a solution of the differential equation y' = cos x. Ans: y = sin x + c Since the constant c is arbitrary, the following are solutions: Thus, y = sin x + c represents the family of solutions of the DE. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz y = sin x + c

21. Families of Solution A solution satisfying a differential equation only differs by a constant for all x in the interval. Assertion: A differential equation of the n th order has an n -parameter family of solution. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

22. Families of Solution Example: The differential equation has the family of solutions UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

23. Families of Solution Challenge Question: Find the family of solutions of the differential equation y' = 2 y UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

24. Families of Solution Example: Consider the second-order differential equation Family of solution: UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

25. General and Particular Solution General solution is a solution containing at least one arbitrary constant. Represents the family of solutions. A solution is said to be particular if it does not contain any arbitrary constant. Represents a specific member of the the family of solutions. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

26. Explicit and Implicit Solution Recall that a solution of a DE is a function. An explicit solution is a function in the form y = f ( x ). An implicit solution (general form) is a function in the form F ( x , y , c 1 , c 2 ,... c n ) = 0 where c 1 , c 2 ,... c n are constants. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

27. Verifying an implicit solution Example: Test whether is a solution of the differential equation on the interval -5 < x < 5. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

28. Deter mination of a particular solution from the general solution: 1. The arbitrary constant, c , must be evaluated 2. How? - there must be given an initial or boundary conditions A problem involving initial condition is called Initial Value Problem. A problem involving boundary condition is called Boundary Value Problem. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

29. Initial Condition - applicable when the independent variable is time , denoted by t , and the conditions are given at t = 0 (start of analysis). Example: Initial velocity of a car Initial charge of a capacitor Initial concentration of a solution Initial population Boundary Condition - applicable if the independent variable is space , denoted by the coordinates x , y , or z , and the conditions are known at any of these coordinates. Example: Electric potential field between two concentric conductors Deformation of a beam at the endpoints UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

30. Initial Value Problem: Using differential calculus, find a solution of the differential equation y' = cos x. Assume y (0) = 2. Ans: y = sin x + 2 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz y = sin x + 2

31. Initial Value Problem: Using differential calculus, find a solution of the differential equation y'' = 12 x 2 . Assume y (0) = 2, y' (0) = 3. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

32. 1st-order Ordinary Differential Equation A DE containing only the first-order derivative y' and may contain y and x . Variable Separable Homogeneous Coefficients Exact Non-exact Linear Bernoulli UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz TODAY! TOMORROW

33. Implicit form: Ex: Review: Explicit and Implicit Form of a Function Explicit form: Ex: UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

34. The standard form is By letting , standard form can be written as Standard and Differential Form of First-Order Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Differential Form

35. A DE in the form is said to be variable separable if the variables can be separated and reduced to the form Test for variable separability: 1. M ( x , y ) can be factored into h ( x ) w ( y ), and 2. N ( x , y ) can be factored into g ( x ) v ( y ) Variable Separable Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

36. Example: Determine if the following DE is variable separable or not 1. Variable Separable Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

37. Example: Determine if the following DE is variable separable or not 2 . Variable Separable Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

38. Example: Determine if the following DE is variable separable or not 3 . Variable Separable Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Homework!

39. Example: Determine if the following DE is variable separable or not 4. Variable Separable Differential Equation UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Homework!

40. Steps to get a solution of a variable separable DE: Step 1: Express the DE in the form Step 2: Separate the independent ( x ) and the dependent variable ( y ) and express in the form Step 3: Integrate both sides of the equation Step 4: Simplify Variable Separable DE UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

41. Solve the following variable separable DE's. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

42. Solve the following variable separable DE's. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

43. Solve the following variable separable DE's. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Homework!

44. Initial Value Problem: Find the particular solution satisfying the initial condition of the variable separable DE below. Variable Separable DE UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Remember: Finding a particular solution to an n-order DE requires n initial values, from y to y (n-1) .

45. Homogeneous Function Definition: A function f ( x , y ) is said to be homogeneous of order n if where r > 0 and n is a constant. 1st-Order DE with Homogeneous Coefficients UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

46. Check the homogeneity of the functions below. If homogeneous, determine its order of homogeneity. 1 . 1st-Order DE with Homogeneous Coefficients UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

47. Check the homogeneity of the functions below. If homogeneous, determine its order of homogeneity. 2. 1st-Order DE with Homogeneous Coefficients UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

48. Check the homogeneity of the functions below. If homogeneous, determine its order of homogeneity. 3 . 1st-Order DE with Homogeneous Coefficients UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Homework!

49. A DE in the form is a first order DE with homogeneous coefficients if M ( x , y ) and N ( x , y ) are each homogeneous functions of the same order n . 1st-Order DE with Homogeneous Coefficients UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

50. Steps to get a solution Step 1: Express the DE into the form Step 2: Use either of the subsitutions y = ux or x = vy . If N ( x , y ) is simpler than M ( x , y ), y = ux is preferable. Step 3: The DE will be reduced into a variable separable DE. Step 4: Do steps in solving variable separable DE's. 1st-Order DE with Homogeneous Coefficients UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

51. Find the solution of the following homogeneous DE. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

52. Find the solution of the following homogeneous DE. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz

53. Find the solution of the following homogeneous DE. UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES - CDO ES 208 Engr. Villaruz Homework!