Mastering Ordinary Differential Equations: A Fun and Simple Guide for Beginners
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Course
Math 202 (YES)
Institution
Forman Christian College
Book
Applied Differential Equations
These detailed course notes are crafted to make learning Ordinary Differential Equations (ODEs) both accessible and engaging for beginners. Each topic is thoroughly explained, guiding you through essential concepts and techniques in a clear, step-by-step manner. Organized to break down complex topi...
ORDINARY DIFFERENTIAL EQUATIONS
LECTURE NO.1
TEXTBOOK: Applied Differential Equations
Murray R. Spiegel (2nd Edition),
Prentice-Hall Mathematics Series (USA).
FREE DOWNLOADABLE FROM
http://93.174.95.29/main/727D4A97DACCA9A2946105037F0A7875
TOPICS TO BE
COVERED
Definitions and examples, formation of ordinary
differential equations, solutions of first order ODEs
(separable, homogeneous, exact, and linear). Solutions
of higher order ODEs (homogeneous and non-
homogenous equations) using method of
undetermined coefficients, variation of parameters
and Laplace transformations.
1
, In y f ( x ), y is a dependent variable
and x is an independent variable.
DIFFERENTIAL EQUATION: An EQUATION which involves
(INCLUDES) DERIVATIVES of a DEPENDENT VARIABLE
WITH RESPECT to ONE or MORE INDEPENDENT
VARIABLES. (PAGE 4)
EXAMPLES
:
2
dy // d dy d y
(1) y y y e , where y and y 2 .
// / x /
dx dx dx dx
2
, y y y e
// / x
is an EQUATION which INCLUDES the DEPENDENT
VARIABLE y, the INDEPENDENT VARIABLE x, and
DERIVATIVES of y with respect to x. Therefore, it is a
DIFFERENTIAL EQUATION.
2 5 2
d s
4
d s
2
ds
( 2 ) 4 2 2 0
dt dt dt
INCLUDES the DERIVATIVES of the DEPENDENT
VARIABLE s with respect to INDEPENDENT VARIABLE
t. Therefore, it is a DIFFERENTIAL EQUATION.
3
, 2V 2V 2V
( 3 ) 0
x 2
y 2
z 2
INCLUDES the DERIVATIVES (PARTIAL DERIVATIVES)
of the DEPENDENT VARIABLE V with respect to
INDEPENDENT VARIABLES x, y and z. Therefore, it is a
DIFFERENTIAL EQUATION.
PARTIAL DERIVATIVE: A derivative of a function of two or
more variables with respect to one variable, the other(s)
being treated as constant.
V
Examples : (1) V x yz
2
2 xyz.
x
Here Partial Derivative is taken with
respect to x keeping y and z constants. 4
, V
( 2) V x yz
2
x z.
2
y
Here Partial Derivative is taken with
respect to y keeping x and z constants.
V
(3) V x yz
2
x y.
2
z
Here Partial Derivative is taken with
respect to z keeping x and y constants.
DIFFERENTIAL EQUATIONS
,ORDINARY DIFFERENTIAL EQUATION: An EQUATION
which involves (INCLUDES) DERIVATIVES WITH
RESPECT to SINGLE (ONLY ONE) INDEPENDENT
VARIABLE. (PAGE 5 )
EXAMPLES: THE DIFFERENTIAL EQUATIONS (1) AND (2)
FROM SLIDES (1) AND (2) ARE ORDINARY DIFFERENTIAL
EQUATIONS AND ARE GIVEN BY
2
dy d
dy d y
(1) y y y e , where y and y 2 .
// / x / //
,PARTIAL DIFFERENTIAL EQUATION: THIS INVOLVES
(INCLUDES) PARTIAL DERIVATIVES WITH RESPECT to
TWO or MORE INDEPENDENT VARIABLES. (PAGE 6)
EXAMPLE: THE DIFFERENTIAL EQUATION (3) FROM
SLIDES (3) IS A PARTIAL DIFFERENTIAL EQUATION
AND IS GIVEN BY
NOTE
IN THIS COURSE WE SHALL STUDY ORDINARY
DIFFERENTIAL EQUATIONS ONLY BUT NOT PARTIAL
DIFFERENTIAL EQUATIONS.
7
, Nth ORDER DERIVATIVE: THE Nth DERIVATIVE OF A
DEPENDENT VARIABLE WITH RESPECT TO AN
INDEPENDENT VARIABLE IS CALLED DERIVATIVE OF
ORDER N OR SIMPLY Nth ORDER DERIVATIVE. (PAGE 6)
dny
For y f ( x ) , n is the nth order derivative
dx
or derivative of order n of y with respect to x.
dy
is first order derivative of y with respect to x.
dx
3
d y
is third order derivative of y with respect to x.
3
dx 8
, CLASSIFICATION of
ORDINARY DIFFERENTIAL EQUATIONS: (Page 6)
HERE CLASSIFICATION MEANS TO ARRANGE ORDINARY
DIFFERENTIAL EQUATIONS IN CLASSES OR CATEGORIES
ACCORDING TO SHARED CHARACTERISTICS OR
QUALITIES.
ORDER of
ORDINARY DIFFERENTIAL EQUATIONS: (Page 6)
THIS IS THE ORDER OF THE HIGHEST DERIVATIVE
WHICH IS PRESENT IN THE ORDINARY
DIFFERENTIAL EQUATION.
SEE EXAMPLES ON NEXT SLIDE
9
, EXAMPLES:
(1) y y y e
// / x
IS AN ORDINARY DIFFERENTIAL EQUATION (ODE)
OF ORDER 2 AS HIGHEST DERIVATIVE PRESENT
IS OF SECOND ORDER DUE TO THE TERM
2
d y
y
//
2
.
dx
10
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