GALGOTIAS COLLEGE OF ENGINEERING & TECHNOLOGY 1, Knowledge Park-II, Greater Noida Department of Applied Science (Mathematics) Major topic-wise PPT Subject Name : Mathematics-II Subject Code : KAS-203 Course Coordinator Dr. Dushyant Kumar (Associate Professor, Mathematics) Faculty Members Dr. Padam Singh Tomar (Asst. Professor, Mathematics) Dr. Vinay Gautam (Asst. Professor, Mathematics)
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Major topic-wise PPT · Major topic-wise PPT Subject Name : Mathematics-II Subject Code : KAS-203 Course Coordinator Dr. Dushyant Kumar (Associate Professor, Mathematics) Faculty
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GALGOTIAS COLLEGE OF ENGINEERING & TECHNOLOGY
1, Knowledge Park-II, Greater Noida
Department of Applied Science (Mathematics)
Major topic-wise PPT
Subject Name : Mathematics-II Subject Code : KAS-203
Course Coordinator Dr. Dushyant Kumar
(Associate Professor, Mathematics)
Faculty Members
Dr. Padam Singh Tomar (Asst. Professor, Mathematics)
Dr. Vinay Gautam (Asst. Professor, Mathematics)
Galgotias College of Engineering and Technology1
Galgotias College of Engineering and Technology2
Ordinary Linear Differential Equations of Second Order
• Linear differential equation of Second order with constant Coefficient
• Linear differential equation of Second orderwith variable coefficient
INTRODUCTION
Galgotias College of Engineering and Technology3
General Form of Linear Differential Equation
Where P and Q both are constants
R is the function of (independent variable)or zero or constant
2
2
d y d yP Q y Rd x d x
x
• Solution of linear Differential Equation of SecondOrder with Constant Coefficient
Galgotias College of Engineering and Technology4
2
2
d y dyP Qy Rdx dx
Now we can write as above D.E2D y PDy Qy R
2D PD Q y R
Galgotias College of Engineering and Technology5
Complete (General) Solution of Differential Equation
General Sol.= Complementary Function + Particular Integral
Y = C.F. + P.I.
2
2
d y dyP Qy Rdx dx
Method of Complementary Function (C.F.)
Galgotias College of Engineering and Technology
6
2 .........(1)D PD Q y R
Replace D by m in Equation (1)
Auxiliary Equation (A.E.)2 0m Pm Q
Case-I when the roots of A.E. are real and distinct
Let
Let
1 2,m m m
Galgotias College of Engineering and Technology7
ThenWhere C1 and C2 are arbitrary Constants.
Example: Solve the given differential equation
Solution: The given equation is
Auxiliary Equation
Where C1 and C2 are arbitrary Constants.
Galgotias College of Engineering and Technology8
Here Particular Integral = 0 (because R=0)
Now General solution Y = C.F. +P.I.
Where C1 and C2 are arbitrary Constants.
Galgotias College of Engineering and Technology9
Example: solve the given differential equation2
2 3 5 4 0d y d y yd x d x
Solution: The given equation is
A.E.Or
Now General Solution Y = C.F. + P.I.
Where C1 and C2 are arbitrary Constants.
Galgotias College of Engineering and Technology10
Case-II When the roots of A.E. are real and equal
Let
Where C1 and C2 are arbitrary Constants.
Then
Example: Solve the given differential equation
Solution: The given equation is
Galgotias College of Engineering and Technology11
Here Auxiliary Equation is
Or
General Solution Y = C.F. + P.I
Where C1 and C2 are arbitrary Constants
Galgotias College of Engineering and Technology12
Case-III when the roots of A.E. are Imaginary (Complex)
Where C1 and C2 are arbitrary Constants
Let
Then
If
Then
Galgotias College of Engineering and Technology13
Example: solve the given differential equation
Solution: The given equation is
Auxiliary Equation is
Now General Solution Y = C.F. + P.I.
Where C1 and C2 are arbitrary Constants.
Galgotias College of Engineering and Technology14
Example-: Solve the given differential equation
Solution: The given equation isHere A.E. is
Now General Solution Y= C.F. + P.I.
Where C1 and C2 are arbitrary constants.
INTRODUCTIONThe series solution method can be defined in two ways:
• Power Series method
• General Series Solution method (Frobenius method)
General form of linear differential equations of Secondorder with variable coefficients:
We can reduce the equation (1) in the given form
Where P and Q both are variable
Ordinary and Singular Points
Let the differential equation
Compare and we get
Put
IfThen is an ordinary point of the differential equation (2)
Thus we can say both of the functions are analytic.
Then is not an ordinary point of the D.E. (2) Therefore is a singular point of the D.E.(2)
Types of Singular Points
• Regular Singular Points
• Irregular Singular Points
Regular Singular Points
Therefore x=a is called regular singular points.
Otherwise x=a is called irregular singular points.
Example-1 Prove thatPoints of Following Differential Equation.
Singular
Also show that is a regular singular point of D.E.
Solution: The given differential equation
is a regular singular point of the above D.E.
Example-2 Prove that is an ordinary point of
following differential equation.
Solution: The given differential equation
is an ordinary point of above D.E.
Series solution of differential equation whenis an ordinary point of the given differential equation
Problem-1 Solve in Series the differential equation about
Solution: The given differential equation
Hence x=0 is an ordinary point of above differential equation
Let the solution of the differential equation in the form
Put the value in D.E., now we get
Now equating the coefficient of
summation and
Now equating the coefficient of
Put Summation &
Now equating the coefficient of
Now we get recurrence relation
Substitute these values in equation (1) now get the solution
Where is called arbitrary constants
Periodic Functions and Fourier Series
Periodic Functions
A function f is periodic
if it is defined for all real and if there is some positive number,
T such that fTf .
Fourier Series
f be a periodic function with period 2The function can be represented by a trigonometric series as:
11
0 sincosn
nn
n nbnaaf
11
0 sincosn
nn
n nbnaaf
What kind of trigonometric (series) functions are we talking about?
32and32
sin,sin,sincos,cos,cos
We want to determine the coefficients,
na and nb .
Let us first remember some useful integrations.
dmndmn
dmn
cos21cos
21
coscos
0
dmn coscos mn
dmn coscos mn
dmndmn
dmn
sin21sin
21
cossin
0
dmn cossin
for all values of m.
dmndmn
dmn
cos21cos
21
sinsin
0
dmn sinsin mn
dmn sinsin mn
Determine 0aIntegrate both sides of (1) from
to
dnbnaa
df
nn
nn
110 sincos
dnb
dnada
df
nn
nn
1
10
sin
cos
000
dadf
002 0
adf
dfa
21
0
0a is the average (dc) value of the
function, f .
dnbnaa
df
nn
nn
2
011
0
2
0
sincos
It is alright as long as the integration is performed over one period.
You may integrate both sides of (1) from
0 to 2 instead.
dnb
dnada
df
nn
nn
2
01
2
01
2
0 0
2
0
sin
cos
002
0 0
2
0
dadf
002 0
2
0 adf
dfa
2
00 21
Determine naMultiply (1) by mcosand then Integrate both sides from
to
dmnbnaa
dmf
nn
nn
cossincos
cos
110
Let us do the integration on the right-hand-side one term at a time.
First term,
00 dma cos
Second term,
dmnan
n coscos1
m
nn admna
coscos1
Second term,
Third term,
01
dmcosnsinb
nn
Therefore,
madmf cos
,2,1cos1 mdmfam
Determine nbMultiply (1) by msinand then Integrate both sides from
to
dmsinnsinbncosaa
dmsinf
nn
nn
110
Let us do the integration on the right-hand-side one term at a time.
First term,
00 dmsina
Second term,
dmsinncosa
nn
1
01
dmsinncosa
nn
Second term,
Third term,
m
nn bdmnb
sinsin1
Therefore,
mbdmsinf
,,mdmsinfbm 211
dfa
21
0
The coefficients are:
,2,1cos1 mdmfam
,2,1sin1 mdmfbm
dfa
21
0
We can write n in place of m:
,,ndncosfan 211
,,ndnsinfbn 211
The integrations can be performed from
0 to 2 instead.
dfa
2
00 21
,,ndncosfan 211 2
0
,,ndnsinfbn 211 2
0
Example 1. Find the Fourier series of the following periodic function.
2
0whenA
whenAf
ff 2
021
2121
2
0
2
0
2
00
dAdA
dfdf
dfa
011
1
1
2
0
2
0
2
0
nnsinA
nnsinA
dncosAdncosA
dncosfan
ncosncoscosncosnA
nncosA
nncosA
dnsinAdnsinA
dnsinfbn
20
11
1
1
2
0
2
0
2
0
oddisnwhen4
1111
20
nA
nA
ncosncoscosncosnAbn
evenisnwhen0
1111
20
nA
ncosncoscosncosnAbn
Therefore, the corresponding Fourier series is
7sin
715sin
513sin
31sin4A
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida
Galgotias College of Engineering and Technology, Greater Noida