1-introduction to ode
TRANSCRIPT
Introduction to Differential Equations
Chapter 1
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Overview
II. Classification of Solutions
Chapter 1: Introduction to Differential Equations
I. Definitions
III. Initial and Boundary Value Problems
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I. Definitions
Learning Objective
AtAt thethe endend ofof thethe section,section, youyou shouldshould bebe ableable totodefinedefine aa differentialdifferential equationequation andand bebe ableable totoclassifyclassify differentialdifferential equationsequations byby type,type, orderorder andandlinearitylinearity..
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I. Definitions
Basic Example
ConsiderConsiderxexf 2)(
xexf 2' 2)(
022)(2)( 22' xx eexfxf
satisfiessatisfies thethe DifferentialDifferential EquationEquation::f
02' yy
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I. Definitions
What is a Differential Equation
AA differentialdifferential equationequation (DE)(DE) isis anan equationequation containingcontaining thethederivativesderivatives ofof oneone oror moremore dependentdependent variablesvariables withwithrespectrespect toto oneone oror moremore independentindependent variablesvariables..
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I. Definitions
Examples
023 )1 '' yy
yxdt
dy
dt
dx423 )3
1 )2 32 xyxdx
dyx
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I. Definitions
Classification
DifferentialDifferential equationsequations (DE)(DE) cancan bebe classifiedclassified byby::
•• TYPETYPE
•• ORDERORDER
•• LINEARITYLINEARITY..
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I. Definitions
Classification by Type
TwoTwo typestypes ofof DifferentialDifferential equationsequations (DE)(DE) existexist::
•• ORDINARYORDINARY DIFFERENTIALDIFFERENTIAL EQUATIONEQUATION (ODE)(ODE)..
AnAn equationequation containingcontaining onlyonly ordinaryordinary derivativesderivatives ofof oneoneoror moremore dependentdependent variablesvariables withwith respectrespect toto aa SINGLESINGLEindependentindependent variablevariable isis saidsaid toto bebe anan OrdinaryOrdinaryDifferentialDifferential EquationEquation (ODE)(ODE)..
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I. Definitions
Examples of ODE
xeydx
dy 5 )1
06 )22
2
ydx
dy
dx
yd
yxdt
dy
dt
dx 2 )3
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I. Definitions
•• PARTIALPARTIAL DIFFERENTIALDIFFERENTIAL EQUATIONSEQUATIONS (PDE)(PDE)..
AnAn equationequation containingcontaining partialpartial derivativesderivatives ofof oneone oror moremoredependentdependent variablesvariables withwith respectrespect toto twotwo oror moremoreindependentindependent variablesvariables isis saidsaid toto bebe aa PartialPartial DifferentialDifferentialEquationEquation (PDE)(PDE)..
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I. Definitions
Examples of PDE
0 )12
2
2
2
y
u
x
u
t
u
t
u
x
u
2 )2
2
2
2
2
x
v
y
u
)3
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I. Definitions
Classification by Order
TheThe orderorder ofof aa differentialdifferential equationequation (ODE(ODE oror PDE)PDE)isis thethe orderorder ofof thethe highesthighest derivativederivative inin thetheequationequation..
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I. Definitions
Examples of Orders
xeydx
dy 35
062
2
ydx
dy
dx
yd
xeydx
dy
dx
yd
45
3
2
2
is of order 1 (or first-order)
is of order 2
is of order 2
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I. Definitions
FirstFirst--orderorder ODEODE areare occasionallyoccasionally writtenwritten inin differentialdifferentialformform ::
Remarks
0),(),( dyyxNdxyxM
AnAn nnthth--orderorder ODEODE inin oneone dependentdependent variablevariable cancan bebeexpressedexpressed inin thethe generalgeneral formform ::
0),...,,,,( )(,,, nyyyyxF
where F is a real-valued function of n+2 variables.
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I. Definitions
Classification by Linearity
AnAn nthnth--orderorder ODEODE
is said to be linear if F is a linear function of the variables:
0),...,,,,( )(,,, nyyyyxF
xgyxayxa...yxayxa n
n
n
n
01
1
1
)(,,, ,...,,, nyyyy
Thus,Thus, thethe generalgeneral formform forfor anan nthnth--orderorder ODEODE is:
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I. Definitions
Classification by Linearity
LinearityLinearity isis characterizedcharacterized byby thethe followingfollowing twotwo propertiesproperties::
1.1. TheThe dependentdependent variablevariable (in(in thisthis casecase )) andand allall itsitsderivativesderivatives mustmust bebe ofof powerpower atat mostmost 11
1. The coefficients for each dependent variable and allits derivatives are only in terms of the independentvariable (in this case ).
y
x
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I. Definitions
Examples for linear ODEs
04 )1 dyxdxxy
02 )2 yyy
xeydx
dyx
dx
yd 5 )3
3
3
xyyx 4
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I. Definitions
Examples for non-linear ODEs
xeyyy- 21 )1
0sin )22
2
ydx
yd
0 )3 2
4
4
ydx
yd
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I. Definitions
Example:
For each of the following ODEs, determine the order and state whether it is linear or non-linear:
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I. Definitions Solution:
ODE Order Linearity
0cos dxxxydy
062
2
dt
dQ
dt
Qd
022
xyyyyxy
0 yyxey
Linear
Linear
Non-linear
Non-linear
1
2
3
2
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I. Definitions Solution:
ODE Order Linearity
sin
012 xdydxy
2
2
2
1
dx
dy
dx
yd
Linear
Non-linear
Non-linear
1
1
2
2sin
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I. Definitions
Exercise-I:
For each of the following ODEs, determine the order and state whether it is linear or non-linear:
tydt
dyt
dt
ydt sin2
2
22
teydt
dyt
dt
ydy
2
22 )1(
12
2
3
3
4
4
ydt
dy
dt
yd
dt
yd
dt
yd
02 tydt
dy
tytdt
ydsin)sin(
2
2
32
3
3
)(cos tytdt
dyt
dt
yd
ttydt
dyt tan12
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II. Classification of Solutions
Learning Objective
At the end of this section, you should be able toAt the end of this section, you should be able to
•• verify the solutions to a given ODE verify the solutions to a given ODE •• identify the different types of solutions of an identify the different types of solutions of an
ODE.ODE.
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Definition:
A solution of a DE is a function that satisfies the DEidentically for all in an interval , where is theindependent variable.
yx I x
II. Classification of Solutions
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Example
xy ln
),0( I
is a solution of the DE: 0'" yxy
xy lnx
y1
'2
1"
xy
xxxyxy
1)
1('''
2
Indeed,
II. Classification of Solutions
011xx
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Definition:
A solution in which the dependent variable is expressedsolely in terms of the independent variable and constantsis said to be an explicit solution.
II. Classification of Solutions
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Definition:
A solution in which the dependent and the independentvariables are mixed in an equation is said to be an implicitsolution.
II. Classification of Solutions
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Examples:
xy ln is an explicit solution of the DE: 0'" yxy
922 yx
922 yx
is an implicit solution of the DE: 0' xyy
Indeed:
Implicit differentiation: 0'22 yyx
0' yyx
1)
2)
II. Classification of Solutions
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General or Particular solution
Example:
Consider the ODE: 0' yyxey is a solution (particular)
xcey (where c is a constant) is a solution (general)
II. Classification of Solutions
xey 2 is also a solution (particular)
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General or Particular solution
• A solution of a DE that is free of arbitrary parameters is called a particular solution.
II. Classification of Solutions
Definitions:
• A solution of a DE representing all possible solutions iscalled a general solution.
• A solution of a DE containing n arbitrary constants is called an n-parameter family of solutions.
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Example
II. Classification of Solutions
xcey is a 1-parameter family of solutions of the DE
0' yy
xx decey is a 2-parameter family of solutions of the DE
0" yy
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Example:
Verify that the indicated function is an explicit solution ofthe given DE :
II. Classification of Solutions
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Example:
1)
II. Classification of Solutions
2;02x
eyyy
2
x
ey
2
2
1'
x
ey
22 )2
1(2'2
xx
eeyy
022 xx
ee
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Example:
2)
II. Classification of Solutions
tey;ydt
dy 20
5
6
5
62420
tey 20
5
6
5
6 te
dt
dyy 20)
5
6(20' te 2024
tt eey
dt
dy 2020
5
6
5
6202420
tt ee 2020 242424 24
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3)
II. Classification of Solutions
xcosey;yyy x 20136 3
xey x 2cos3 xexey xx 2sin22cos3' 33 xey x 2sin23 3
xexeyy xx 2cos42sin6'3" 33 yxey x 42sin6'3 3
yyy 136 yyyxey x 13'642sin6'3 3
yxexey xx 92sin62sin233 33
yxey x 92sin6'3 3
092sin62sin69 33 yxexey xx
Example:
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4)
II. Classification of Solutions
xtanxseclnxcosy;xtanyy
xxxy tanseclncos
xxxxxy seccostanseclnsin' 1tanseclnsin xxx
xxxxxy secsintanseclncos" xxxx tantanseclncos
xxxxxxxyy tanseclncostantanseclncos
xtan
Example:
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5)
II. Classification of Solutions
t
t
ec
ecPPPP
1
1
1;1
t
t
ec
ecP
1
1
1
21
1111
1
1'
t
tttt
ec
ececececP
21
12
1
221
2211
11 t
t
t
ttt
ec
ec
ec
ececec
t
t
t
t
ec
ec
ec
ecPP
1
1
1
1
11
11
t
tt
t
t
ec
ecec
ec
ec
1
11
1
1
1
1
1
'
12
1
1 Pec
ect
t
Example:
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6)
II. Classification of Solutions
xx xececy;ydx
dy
dx
yd 2
2
2
12
2
044
xxx xeececy 222
21 22' xxx xececec 2
22
22
1 22
xxxx ecyecxecec 22
22
22
21 22
xecyy 222'2"
yyy 4'4" yyecy x 4'42'2 22 yyec x 4'22 2
2
04222 22
22 yecyec xx
Example:
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7)
II. Classification of Solutions
0
02
2
x,x
x,xy,02 yyx
0,2
0,,2'
xx
xxy
0,022
0,0)(222'
22
22
xxx
xxxyxy
Solution on ,
0limlim)0()0(
lim0
2
00
h
h
h
h
yhy
hhh
0limlim)0()0(
lim0
2
00
h
h
h
h
yhy
hhh
0)0(' y
02 yyx is still valid at 0.
Example:
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Exercise-II:
Verify that the indicated functions are explicit solutions ofthe given DE :
II. Classification of Solutions
tttttyttyy
tttyttytytyyt
ttyttytytyyt
tety
ttytyyy
ttytyyt
t
sincosln)(cos)( ;2
0 ,sec'')5
ln)( ,)( ;0 ,04'5'')4
)( ,)( ;0 ,0'3''2)3
3)( ,
3)( ,34)2
3 ,)1
22
21
2
12
21
12
21)3()4(
22
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III. Initial and Boundary Value Problems
Learning Objective
At the end of this section, you should and be able toAt the end of this section, you should and be able to
•• Define IVP and BVP Define IVP and BVP •• Verify solutions to DE subjected to given initial Verify solutions to DE subjected to given initial
conditions.conditions.
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III. Initial and Boundary Value Problems
Definition
A DE with initial conditions on the unknown function and itsA DE with initial conditions on the unknown function and itsderivatives, all given at the same value of the independentderivatives, all given at the same value of the independentvariable, is called an variable, is called an initialinitial--value problemvalue problem, IVP., IVP.
0x
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III. Initial and Boundary Value Problems
Examples
3)0( ,0 )1 yyy
25)1(' ,0 )2 yyy
5)2( ,0'2'' )3 yyyy
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III. Initial and Boundary Value Problems
Definition
A DE with initial conditions on the unknown function and itsA DE with initial conditions on the unknown function and itsderivatives, all given at different values (e.g. at and )derivatives, all given at different values (e.g. at and )of the independent variable, is called an of the independent variable, is called an boundaryboundary--valuevalueproblemproblem, BVP., BVP.
0x 1x
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III. Initial and Boundary Value Problems
Examples
22
,1;2 )1
yyeyy x
11,10;2 )2 yyeyy x
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III. Initial and Boundary Value Problems
Examples
Find the solution of the IVP or BVP if the general solution is the Given one:
,23;0 )1 yyy xecxy 1
313 ecy 23 y
231 ec
31 2ec
xx eeexy 33 22solution of the IVP:
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III. Initial and Boundary Value Problems
16
,08
;04 )2
yyyy xcxcxy 2cos2sin 21
82cos
82sin
821
ccy
2221 cc
08
y 21 cc
Examples
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III. Initial and Boundary Value Problems
solution of the BVP:
62cos
62sin
621
ccy
22
3 21 cc
16
y 1
2
3 21 cc
23 11 cc
13
21
c
13
22
c
xxy 2cos2sin13
2
Examples
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III. Initial and Boundary Value Problems
xcxcxy 2cos2sin 21
0cos0sin0 21 ccy 2c
10 y 12 c
,22
,10;04 )3
yyyy
cossin2
21 ccy
2c
22
y 22 c
22 c
12 cIMPOSSIBLE NO SOLUTION
Examples
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III. Initial and Boundary Value Problems
Exercise-III
1) Determine and so that will satisfy the conditions :
08
y 2
8
y
1c 2c 12cos2sin 21 xcxcxy
2) Determine and so that will satisfy the conditions :
xececxy xx sin222
1 1c 2c
00 y 10 y
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End Chapter 1
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I. Definitions Solution-I:
ODE Order Linearity
Linear
Non-Linear
Linear
Non-linear
2
2
4
1
tydt
dyt
dt
ydt sin2
2
22
teydt
dyt
dt
ydy
2
22 )1(
12
2
3
3
4
4
ydt
dy
dt
yd
dt
yd
dt
yd
02 tydt
dy
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I. Definitions Solution-I:
ODE Order Linearity
Non-linear
Linear
Linear
2
3
1
tytdt
ydsin)sin(
2
2
32
3
3
)(cos tytdt
dyt
dt
yd
ttydt
dyt tan12
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Solution-II:
II. Classification of Solutions
0
2323
)23()23('
23'
3 , )1
22
2
22
tttt
ttttyty
ty
ttytyyt
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Solution-II:
II. Classification of Solutions
tteeetytyty
etyety
etyetyt
ety
tt
tytyty
tytytytyt
ty
tety
ttytyyy
ttt
tt
ttt
t
34)(3)(4)(
)( ,)(
,)( ,3
1)( ,
3)(
33)(3)(4)(
0)()()( ,3
1)( ,
3)(
3)( ,
3)( ,34 )2
2)3(
2)4(
2
)4(2
)3(2
"2
'22
1)3(
1)4(
1
)4(1
)3(1
''1
'11
21)3()4(
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Solution-II:
II. Classification of Solutions
034
)(3)2(232
2)(,)(
02
3
2
1
)2
1(3)
4
1(232
4
1)(,
2
1)(
)( ,)( ;0 ,0'3''2 )3
111
1232'1
"2
2
3''2
2'2
21
21
21
21
21
23
2'1
"1
2
23
''1
21
'1
12
21
12
ttt
tttttytyyt
ttytty
ttt
tttttytyyt
ttytty
ttyttytytyyt
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Solution-II:
II. Classification of Solutions
0ln45ln105ln6
ln4)ln2(5)5ln6(45
32ln6)( ,ln2)( ,ln)(
04106
4)2(5)6(45
6)( ,2)( ,)(
ln)( ,)( ;0 ,04'5'' )4
22222
2334422
'2
"2
2
444"2
33'2
22
222
23421
'1
"1
2
4"1
3'1
21
22
21
2
tttttttt
ttttttttttytyyt
tttttyttttyttty
ttt
tttttytyyt
ttyttytty
tttyttytytyyt
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Solution-II:
II. Classification of Solutions
ttt
ttt
t
t
tttttttt
tttyy
tttt
ttt
tttt
ttttty
tttt
tttt
ttttty
tttttyttyy
seccos
1
cos
cossincos
cos
sin
sincosln)(cossincoscos
sincosln)(cos''
sincoscos
sincosln)(cos
sincos)cos
sin(sincosln)(cos)(''
coscosln)(sin
cossin)cos
sin(coscosln)(sin)('
sincosln)(cos)( ;2
0 ,sec'' )5
222
2
2
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III. Initial and Boundary Value Problems
Solution-III
1)
208
21
ccy
128
21
ccy
12cos2sin 21 xcxcxy
12
2)(1
82cos
82sin
821
21
ccccy
)(28
2sin28
2cos28
' 2121 ccccy
2
212
c
xcxcxy 2sin22cos2' 21
2
211
c
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III. Initial and Boundary Value Problems
Solution-III
2)
1200 ccy
112210 121 cccy
210 ccy
xececxy xx cos22' 22
1
1 ,1 21 cc
xececxy xx sin222
1
220' 21 ccy
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