Mastering Ordinary Differential Equations: A Fun and Simple Guide for Beginners

ORDINARY DIFFERENTIAL EQUATIONS
LECTURE NO.1
TEXTBOOK: Applied Differential Equations
Murray R. Spiegel (2nd Edition),
Prentice-Hall Mathematics Series (USA).
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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 PARTIAL
DIFFERENTIAL DIFFERENTIAL
EQUATIONS EQUATIONS 5

,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 / //

dx dx dx dx
AND
2 5 2
d s
4
 d s
2
  ds 
( 2 )  4   2  2      0
 dt   dt   dt  6

,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

  2V    2V    2V 
         0
 x  y  z
2 2 2
  

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