1-Introduction to ODE

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


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..


I. Definitions

Basic Example

ConsiderConsiderxexf 2)(

xexf 2' 2)(

022)(2)( 22' xx eexfxf

satisfiessatisfies thethe DifferentialDifferential EquationEquation::f

02' yy


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..


I. Definitions

Examples

023 )1 '' yy

yxdt

dy

dt

dx423 )3

1 )2 32 xyxdx

dyx


I. Definitions

Classification

DifferentialDifferential equationsequations (DE)(DE) cancan bebe classifiedclassified byby::

•• TYPETYPE

•• ORDERORDER

•• LINEARITYLINEARITY..


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)..


I. Definitions

Examples of ODE

xeydx

dy 5 )1

06 )22

2

ydx

dy

dx

yd

yxdt

dy

dt

dx 2 )3


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)..


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


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..


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


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.


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:


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


I. Definitions

Examples for linear ODEs

04 )1 dyxdxxy

02 )2 yyy

xeydx

dyx

dx

yd 5 )3

3

3

xyyx 4


I. Definitions

Examples for non-linear ODEs

xeyyy- 21 )1

0sin )22

2

ydx

yd

0 )3 2

4

4

ydx

yd


I. Definitions

Example:

For each of the following ODEs, determine the order and state whether it is linear or non-linear:


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


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


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



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.


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



Example

xy ln

),0( I

is a solution of the DE: 0'" yxy

xy lnx

y1

'2

1"

xy

xxxyxy

1)

1('''

2

Indeed,


011xx


Definition:

A solution in which the dependent variable is expressedsolely in terms of the independent variable and constantsis said to be an explicit solution.



Definition:

A solution in which the dependent and the independentvariables are mixed in an equation is said to be an implicitsolution.



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)



General or Particular solution

Example:

Consider the ODE: 0' yyxey is a solution (particular)

xcey (where c is a constant) is a solution (general)


xey 2 is also a solution (particular)


General or Particular solution

• A solution of a DE that is free of arbitrary parameters is called a particular solution.


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.


Example


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


Example:

Verify that the indicated function is an explicit solution ofthe given DE :



Example:

1)


2;02x

eyyy

2

x

ey

2

2

1'

x

ey

22 )2

1(2'2

xx

eeyy

022 xx

ee


Example:

2)


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


3)


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:


4)


xtanxseclnxcosy;xtanyy

xxxy tanseclncos

xxxxxy seccostanseclnsin' 1tanseclnsin xxx

xxxxxy secsintanseclncos" xxxx tantanseclncos

xxxxxxxyy tanseclncostantanseclncos

xtan

Example:


5)


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:


6)


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:


7)


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:


Exercise-II:

Verify that the indicated functions are explicit solutions ofthe given DE :


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



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.



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



Examples

3)0( ,0 )1 yyy

25)1(' ,0 )2 yyy

5)2( ,0'2'' )3 yyyy



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



Examples

22

,1;2 )1

yyeyy x

11,10;2 )2 yyeyy x



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:



16

,08

;04 )2

yyyy xcxcxy 2cos2sin 21

82cos

82sin

821

ccy

2221 cc

08

y 21 cc

Examples



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



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



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


End Chapter 1


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


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


Solution-II:


0

2323

)23()23('

23'

3 , )1

22

2

22

tttt

ttttyty

ty

ttytyyt


Solution-II:


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(


Solution-II:


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


Solution-II:


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


Solution-II:


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



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



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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1-Introduction to ODE