Week 1 [compatibility mode]
Transcript of Week 1 [compatibility mode]
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Prepared ByAnnie ak Joseph
Prepared ByAnnie ak Joseph Session 2008/2009
KNF1023Engineering
Mathematics II
Introduction to ODEs
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Learning Objectives
Apply an ODEs in real life application
Solve the problems of ODEs
Describe the concept of ODEs
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Introduction to ODEs
Order ofODE
Introductionto ODEs
Solving anODE –general,particular,exactsolutions
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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
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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
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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
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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
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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.
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Solving of an Ordinary DifferentialEquations
Example 1:
06'' xy
Cxxdxdx
dyy 2, 36
DCxxdxCxy 32 )3(
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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.
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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(
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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
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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.
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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(
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Applications of ODEs
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Summary
Order of ODE
Solving an ODE
general, particular, exact solutions
ODEs
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Prepared ByAnnie ak Joseph
Prepared ByAnnie ak Joseph Session 2008/2009