1. Introduction to Ordinary Differential Equations
Chapter 1
Introduction to Ordinary Differential Equations

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

3. Learning Objective I. Definitions
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. Basic Example I. Definitions Consider
satisfies the Differential Equation:

5. What is a Differential Equation
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. Classification I. Definitions
Differential equations (DE) can be classified by:
TYPE
ORDER
LINEARITY.

8. Classification by Type
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. Classification by Order
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. Examples of Orders I. Definitions is of order 1 (or first-order)

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

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

16. Examples for linear ODEs
I. Definitions
Examples for linear ODEs

17. Examples for non-linear ODEs
I. Definitions
Examples for non-linear ODEs

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

19. Solution: I. Definitions Order Linearity ODE 1 Linear 2 Linear 31
Linear 2
Linear 3
Non-linear 2
Non-linear

20. Solution: I. Definitions Order Linearity ODE 1 Linear 1 Non-linear 21
Linear 1
Non-linear 2
Non-linear

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

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

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

25. Definition: II. Classification of Solutions
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. Definition: II. Classification of Solutions
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. Example: II. Classification of Solutions
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. Exercise-II: II. Classification of Solutions
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
IMPOSSIBLE
NO SOLUTION

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