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Prepared ByAnnie ak Joseph
Prepared ByAnnie ak Joseph Session 2008/2009
KNF1023Engineering
Mathematics II
Introduction to ODEs
Learning Objectives
Apply an ODEs in real life application
Solve the problems of ODEs
Describe the concept of ODEs
Introduction to ODEs
Order ofODE
Introductionto ODEs
Solving anODE –general,particular,exactsolutions
Basic Concept
An ordinary differential equation is anequation with relationship betweendependent variable (“y”), independentvariable (“x”) and one or morederivative of with respect to .
Example:1.
2.
3.
y x
45, xy
8,, xyy
xyxyyyx ,,,,,, 31022
Basic Concept
Ordinary Differential equations differentfrom partial differential equations
Partial Differential equations-> unknownfunction depends on two or morevariables, so that they are morecomplicated
02
2
2
2
dy
Vd
dx
Vd
Order of ODEs:
The order of a differential equation is theorder of the highest derivative involvedin the equation.
Example:
1.2.3.4.
xy cos, 04,, yy
22,,,,,,2 )2(2 yxyeyyx x xyxyyyx ,,,,,, 31022
Arbitrary Constants
An arbitrary constant, often denoted by aletter at the beginning of the alphabetsuch as A, B,C, , etc. may assumevalues independently of the variablesinvolved. For example in , c1and c2 are arbitrary constants.
212 cxcxy
21 , cc
Solving of an Ordinary DifferentialEquations
A solution of a differential equation is arelation between the variables which isfree of derivatives and which satisfies thedifferential equation identically.
Solving of an Ordinary DifferentialEquations
Example 1:
06'' xy
Cxxdxdx
dyy 2, 36
DCxxdxCxy 32 )3(
Concept of General Solution
A solution containing a number ofindependent arbitrary constants equal tothe order of the differential equation iscalled the general solution of the equation.
We regard any function y(x) with Narbitrary constants in it to be a generalsolution of N th order ODE in y=y(x) if thefunction satisfies the ODE.
Concept of General Solution
Example 2 : is a solutionfor ODE
DCxxxy 38)(
xy 48''
xdx
ydy 48
2
2,,
Cxxdxy 2' 2448
DCxxdxCxy 32 8)24(
Particular Solution
When specific values are given to at leastone of these arbitrary constants, thesolution is called a particular solution.
Example 3:
Dxxxy 28)( 3
58)( 3 Cxxxy
158)( 3 xxxy
Exact Solution
A solution of an ODE is exact if thesolution can be expressed in terms ofelementary functions.
We regards a function as elementary if itsvalue can be calculated using an ordinaryscientific hand calculator.
Exact Solution
Thus the general solutionof the ODE is exact.
We may not able to find exact solutionfor some ODEs. As example, considerthe ODE
DCxxxy 38)(
xy 48''
dxx
xy
x
x
dx
dy
)sin(
)sin(
Applications of ODEs
Summary
Order of ODE
Solving an ODE
general, particular, exact solutions
ODEs
Prepared ByAnnie ak Joseph
Prepared ByAnnie ak Joseph Session 2008/2009