Introduction to Ordinary Differential Equations

1. Introduction to Ordinary Differential Equations Chapter 1

2. Chapter 1:Introduction to Differential Equations Overview I. Definitions II. Classification of Solutions

3. I. Definitions Learning Objective At the end of the section, you should be able to define a differential equation and classify differential equations by type, order and linearity.

4. I. Definitions Basic Example Consider satisfies the Differential Equation:

5. I. Definitions What is a Differential Equation A differential equation (DE) is an equation containing the derivatives of one or more dependent variables with respect to one or more independent variables.

6. I. Definitions Examples

7. I. Definitions Classification • Differential equations (DE) can be classified by: • TYPE • ORDER • LINEARITY.

8. I. Definitions Classification by Type • Two types of Differential equations (DE) exist: • ORDINARY DIFFERENTIAL EQUATION (ODE). • An equation containing only ordinary derivatives of one or more dependent variables with respect to a SINGLE independent variable is said to be an Ordinary Differential Equation (ODE).

9. I. Definitions Examples of ODE

10. I. Definitions • PARTIAL DIFFERENTIAL EQUATIONS (PDE). • An equation containing partial derivatives of one or more dependent variables with respect to TWO or more independent variables is said to be a Partial Differential Equation (PDE).

11. I. Definitions Examples of PDE

12. I. Definitions Classification by Order The order of a differential equation (ODE or PDE) is the order of the highest derivative in the equation.

13. I. Definitions Examples of Orders is of order 1 (or first-order) is of order 2 is of order 2

14. I. Definitions Remark First-order ODEs are occasionally written in differential form :

15. I. Definitions Classification by Linearity The general form for an nth-order ODEis: The general form for an 2nd-order ODEis:

16. I. Definitions Examples for linear ODEs

17. I. Definitions Examples for non-linear ODEs

18. I. Definitions Example: For each of the following ODEs, determine the order and state whether it is linear or non-linear:

19. I. Definitions Solution: 1 Linear 2 Linear 3 Non-linear 2 Non-linear

20. I. Definitions Solution: 1 Linear 1 Non-linear 2 Non-linear

21. I. Definitions Exercise-I: For each of the following ODEs, determine the order and state whether it is linear or non-linear:

22. II. Classification of Solutions Learning Objective • At the end of this section, you should be able to • verify the solutions to a given ODE • identify the different types of solutions of an ODE. • Define IVP, BVP

23. II. Classification of Solutions Definition: A solution of a DE is a function that satisfies the DE identically for all in an interval , where is the independent variable.

24. II. Classification of Solutions Example is a solution of the DE: Indeed,

25. II. Classification of Solutions Definition: A solution in which the dependent variable is expressed solely in terms of the independent variable and constants is said to be an explicit solution.

26. II. Classification of Solutions Definition: A solution in which the dependent and the independent variables are mixed in an equation is said to be an implicit solution.

27. II. Classification of Solutions Examples: 1) is an explicit solution of the DE: 2) is an implicit solution of the DE: Indeed: Implicit differentiation:

28. II. Classification of Solutions General or Particular solution Example: Consider the ODE: is a solution (particular) is also a solution (particular) (where c is a constant) is a solution (general)

29. II. Classification of Solutions General or Particular solution Definitions: • A solution of a DE that is free of arbitrary parameters is called a particular solution. • A solution of a DE representing all possible solutions is called a general solution.

30. II. Classification of Solutions Example is a 1-parameter family of solutions of the DE is a 2-parameter family of solutions of the DE

31. II. Classification of Solutions Example: Verify that the indicated function is an explicit solution of the given DE :

32. II. Classification of Solutions Example: 1)

33. II. Classification of Solutions Example: 2)

34. II. Classification of Solutions Example: 3)

35. II. Classification of Solutions Example: 4)

36. II. Classification of Solutions Example: 5)

37. II. Classification of Solutions Example: 6)

38. II. Classification of Solutions Exercise-II: Verify if the indicated functions are explicit solutions of the given DE :

39. II. Classification of Solutions Definition A DE with initial conditions on the unknown function and its derivatives, all given at the same value of the independent variable, is called an initial-value problem, IVP.

40. II. Classification of Solutions Examples

41. II. Classification of Solutions Definition A DE with initial conditions on the unknown function and its derivatives, all given at different values (e.g. at and ) of the independent variable, is called a boundary-value problem, BVP.

42. II. Classification of Solutions Examples

43. II. Classification of Solutions Examples Find the solution of the IVP or BVP if the general solution is the given one: solution of the IVP:

44. II. Classification of Solutions Examples

45. II. Classification of Solutions Examples solution of the BVP:

46. II. Classification of Solutions Examples NO SOLUTION IMPOSSIBLE

47. II. Classification of Solutions Exercise-III • Determine and so that will satisfy the conditions : 2) Determine and so that will satisfy the conditions :

48. End Chapter 1