Ordinary Differential equations and their solutions, Lecture 1, O.D.E

Date: 12.02.2011
Course conducted by SJ(Sohana Jahan)
Course Title MTH 203
Course Name: Ordinary Differential Equations

Chapter one:
Differential Equations & their Solutions
In order to know what is Ordinary differential equation we must know Differential equation first.

Definition: An equation involving derivatives of one or more dependent variables with respect to one or more independent variable is called a differential equation.


example:

Note: It is to be noted that, in the above equations two types of derivative operator has been used. It should be minded that d/dx( read d dx of something like y,z etc) is used when one or more dependent variable is differentiate with respect to one independent variable.

other hand,  δ/δx (read δ(del) δx(del x)  of something like y,z etc) is used when dependent variable /variables are differentiate with respect to more than one independent variable .

Ordinary Differential Equation:A differential equation involving ordinary derivatives  of one or more dependent variables with respect to a single independent variable is called an ordinary differential equation.
Example:

Classification of differential equations:

we can classify differential equations in the following three  ways

1) Depending on Independent variable

2) Differential equations are linear of non linear

3) According to the order of the de



1. Depending on Independent variable:  Depending on independent variable we can classify differential equation in two ways

a. Ordinary Differential equations: If the independent variable is single by which respect we differentiate one or more dependent variable then these type of differential equations are ordinary diff. equation

Definition & example: Its definition and example are given before

b. Partial Differential equations: If the independent variable is more than one by which respect we differentiate one or more dependent variable then these type of differential equations are partial diff. equation.

Definition: A differential equation involving partial derivatives of one or more dependent variable with respect to more than one independent variable is called a partial diff. equation.

Example:

2.Differential equations are linear o non linear:

a. Linear ordinary equation(defn): A linear ordinary diff. equation of order n in the dependent variable y and the independent variable x, is an equation that is in, or can be expressed in the form

Note: In the above expression it is to be noted that
 i) the dependent variable y and its various derivatives occur in the first power.
 ii) no products of y and/or any of its derivatives are present
 iii) No transcendental functions of y and/or derivatives occur.

Example:

b. Non linear ordinary differential equation: A non linear ordinary  differential equation is an ordinary differential equation that is not linear.

Example:

3. According to Order of the differential equation:

Definition: The order of the highest ordered derivative involved in a differential equation is called the order of the differential equation.
Example:

 In above picture no. 2 is an ordinary diff. equation of second order.