1. KNF1023
Engineering
Mathematics II
Introduction to ODEs
Prepared By
Annie ak Joseph
Prepared By
Annie ak Joseph Session 2008/2009

2. Learning Objectives
Describe the concept of ODEs
Solve the problems of ODEs
Apply an ODEs in real life application

3. Introduction to ODEs
Introduction
to ODEs
Order of Solving an
ODE ODE –
general,
particular,
exact
solutions

4. Basic Concept
An ordinary differential equation is an
equation with relationship between
dependent variable (“y”), independent
variable (“x”) and one or more
derivative of y with respect to x.
Example:
1. y  5 x 4
,
2. y ,,  xy  8
3. 2 x 2 y 10 y , , , ,  3 xy ,,
 xy

5. Basic Concept
Ordinary Differential equations different
from partial differential equations
Partial Differential equations-> unknown
function depends on two or more
variables, so that they are more
complicated
d 2V d 2V
 2
 2
0
dx dy

6. Order of ODEs:
The order of a differential equation is the
order of the highest derivative involved
in the equation.
 Example:
1. y  cos x
,
2. y ,,  4 y  0
3. 2 x 2 y10 y ,,,,  3 xy ,,  xy
4. x y y  2e y  (x  2) y
2 ,,, , x ,, 2 2

7. Arbitrary Constants
An arbitrary constant, often denoted by a
letter at the beginning of the alphabet
such as A, B,C, c 1 , c 2 , etc. may assume
values independently of the variables
involved. For example in y  x2  c1 x  c2 , c 1
and c2 are arbitrary constants.

8. Solving of an Ordinary Differential
Equations
A solution of a differential equation is a
relation between the variables which is
free of derivatives and which satisfies the
differential equation identically.

9. Solving of an Ordinary Differential
Equations
Example 1:
y  6x  0
''
dy
y 
,
  6 xdx  3 x  C
2
dx
y   (3 x  C ) dx  x  Cx  D
2 3

10. Concept of General Solution
A solution containing a number of
independent arbitrary constants equal to
the order of the differential equation is
called the general solution of the equation.
We regard any function y(x) with N
arbitrary constants in it to be a general
solution of N th order ODE in y=y(x) if the
function satisfies the ODE.

11. Concept of General Solution
 Example 2 : y ( x)  8 x 3  Cx  D is a solution
for ODE y ''  48 x
2
d y
y  2  48 x
,,
dx
y '   48 xdx  24 x 2  C
y   ( 24 x 2  C ) dx  8 x 3  Cx  D

12. Particular Solution
 When specific values are given to at least
one of these arbitrary constants, the
solution is called a particular solution.
 Example 3:
y ( x)  8 x 3  2 x  D
y ( x)  8 x  Cx  5
3
y ( x)  8 x 3  5 x  1

13. Exact Solution
A solution of an ODE is exact if the
solution can be expressed in terms of
elementary functions.
We regards a function as elementary if its
value can be calculated using an ordinary
scientific hand calculator.

14. Exact Solution
Thus the general solution y ( x)  8 x 3  Cx  D
of the ODE y ''  48 x is exact.
We may not able to find exact solution
for some ODEs. As example, consider
the ODE
dy sin( x)

dx x
sin( x)
y dx
x

15. Applications of ODEs

16. Summary
Order of ODE
Solving an ODE ODEs
general, particular, exact solutions

17. Prepared By
Annie ak Joseph
Prepared By
Annie ak Joseph Session 2008/2009