b.tech admissions in delhi

Admission in India 2015

By:

admission.edhole.com

http://school.edhole.com/

Ordinary Differential Equations

S.-Y. LeuSept. 21,28, 2005



1.1 Definit ions and Terminology

1.2 Init ial-Value Problems 1.3 Differential Equation as Mathematical Models

CHAPTER 1Introduction to Differential Equations



DEFINITION: differential equationAn equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a

differential equation (DE(.

)Zill, Definition 1.1, page 6(.




Recall CalculusDefinit ion of a DerivativeIf , the derivative of or

With respect to is defined as

The derivative is also denoted by or


)(xfy = y )(xfx

h

xfhxf

dx

dyh

)()(lim0

−+=→

dx

dfy ,' )(' xf



Recall the Exponential function

dependent variable: y independent variable: x


xexfy 2)( ==

yedx

xde

dx

ed

dx

dy xxx

22)2()( 22

2

==

==



Differential Equation: Equations that involve dependent variables and their derivatives with respect to the independentvariables.

Differential Equations are classified by

type, order and l inearity.




Differential Equations are classified bytype, order and l inearity.

TYPEThere are two main types of differential

equation: “ordinary” and “partial”.




Ordinary differential equation (ODE( Differential equations that involve only ONE independent variable are called ordinary differential equations.

Examples:

, , and

only ordinary (or total ( derivatives


xeydx

dy =+ 5 062

2

=+− ydx

dy

dx

yd yxdt

dy

dt

dx +=+ 2



Partial differential equation (PDE(Differential equations that involve two or more independent variables are called partial differential equations.Examples:

and

only partial derivatives


t

u

t

u

x

u

∂∂−

∂∂=

∂∂

22

2

2

2

x

v

y

u

∂∂−=

∂∂



ORDERThe order of a differential equation is the order of the highest derivative found in the DE.

second order f irst order


xeydx

dy

dx

yd =−

+ 45

3

2

2




xeyxy =− 2'

3'' xy =

0),,( ' =yyxFfirst order

second order 0),,,( ''' =yyyxF

Written in differential form: 0),(),( =+ dyyxNdxyxM



LINEAR or NONLINEAR

An n-th order differential equation is said to be l inear if the function

is linear in the variables

there are no multiplications among dependent variables and their derivatives. All coeff icients are functions of independent variables.

A nonlinear ODE is one that is not linear, i.e. does not have the above form.


)1(' ,..., −nyyy

)()()(...)()( 011

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n =++++ −

−

−

0),......,,( )(' =nyyyxF



LINEAR or NONLINEAR

orlinear first-order ordinary differential equation

linear second-order ordinary differential equation

linear third-order ordinary differential equation


0)(4 =−+ xydx

dyx

02 ''' =+− yyy

04)( =+− xdydxxy

xeydx

dyx

dx

yd =−+ 533

3



LINEAR or NONLINEAR

coefficient depends on ynonlinear first-order ordinary differential equation

nonlinear function of ynonlinear second-order ordinary differential equation

power not 1 nonlinear fourth-order ordinary differential equation


0)sin(2

2

=+ ydx

yd

xeyyy =+− 2)1( '

024

4

=+ ydx

yd



LINEAR or NONLINEAR

NOTE:


...!7!5!3

)sin(753

+−+−= yyyyy ∞<<−∞ x

...!6!4!2

1)cos(642

+−+−= yyyy ∞<<−∞ x



Solutions of ODEsDEFINITION: solution of an ODEAny function , defined on an interval I and possessing at least n derivatives that are continuouson I, which when substituted into an n-th order ODE reduces the equation to an identity, is said to be a

solution of the equation on the interval. )Zill, Definition 1.1, page 8(.


φ



Namely, a solution of an n-th order ODE is a function which possesses at least nderivatives and for which

for all x in I

We say that satisfies the differential equation on I.


0))(),(),(,( )(' =xxxxF nφφφ



Verification of a solution by substitution Example:

left hand side:

right-hand side: 0

The DE possesses the constant y=0 trivial solution


xxxx exeyexey 2, ''' +=+=

xxeyyyy ==+− ;02 '''

0)(2)2(2 ''' =++−+=+− xxxxx xeexeexeyyy



DEFINITION: solution curveA graph of the solution of an ODE is called a solution curve, or an integral curve of

the equation.





DEFINITION: famil ies of solutions A solution containing an arbitrary constant (parameter) represents a set ofsolutions to an ODE called a one-parameter

family of solutions. A solution to an n−th order ODE is a n-parameter family of

solutions .

Since the parameter can be assigned an infinite number of values, an ODE can have an infinite number of solutions.

0),,( =cyxG

0),......,,( )(' =nyyyxF





2' =+ yyxkex −+= 2)(φ

2' =+ yyxkex −+= 2)(φxkex −−=)(φ '

22)(φ)(φ ' =++−=+ −− xx kekexx


Figure 1.1 Integral curves of y’+ y = 2 for k = 0, 3, –3, 6, and –6.

©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

xkey −+= 2




1' +=x

yy

Cxxxx += )ln()(φ 0>x

Cxx ++= 1)ln()(φ '

1)(φ

1)ln(

)(φ ' +=++=x

x

x

Cxxxx

for all ,



Figure 1.2 Integral curves of y’ + ¹ y = e x

for c =0,5,20, -6, and –10. x

)(1

cexex

y xx +−=


Second-Order Differential Equation

Example: is a solution of

By substitution:

)4sin(17)4cos(6)(φ xxx −=016'' =+ xy

0φ16φ

)4sin(272)4cos(96φ

)4cos(68)4sin(24φ

''

''

'

=++−=

−−=xx

xx

0),,,( ''' =yyyxF

( ) 0)(φ),(φ),(φ, ''' =xxxxFadmission.edhole.com

Second-Order Differential EquationConsider the simple, l inear second-order equation

,

To determine C and K, we need two initial conditions, one specify a point lying on the solut ion curve and the

other its slope at that point, e.g. ,

WHY ???

012'' =− xy

xy 12'' = Cxxdxdxxyy +=== ∫∫ 2'' 612)('

KCxxdxCxdxxyy ++=+== ∫∫ 32' 2)6()(

Cy =)0('Ky =)0(


Second-Order Differential Equation

IF only try x=x1, and x=x2

It cannot determine C and K,

xy 12'' =KCxxy ++= 32

KCxxxy

KCxxxy

++=

++=

23

22

13

11

2)(

2)(



Figure 2.1 Graphs of y = 2x³ + C x +K for various values of C and K.

e.g. X=0, y=k



Figure 2.2 Graphs of y = 2x³ + C x + 3 for various values of C.

To satisfy the I.C. y(0)=3The solution curve must pass through (0,3)

Many solut ion curves through (0,3)



Figure 2.3 Graph of y = 2x³ - x + 3.

To satisfy the I.C. y(0)=3,y’(0)=-1, the solution curve must pass through (0,3)having slope -1


Solutions General Solution: Solutions obtained from integrating the differential equations are called general solutions. The general solution of a nth order ordinary differential equation contains n arbitrary constants resulting from

integrating times. Particular Solution: Particular solutions are the solutions obtained by assigning specific values to the

arbitrary constants in the general solutions. Singular Solutions: Solutions that can not be expressed by the general solutions are called singular

solutions.



DEFINITION: implicit solutionA relation is said to be an implicit solution of an ODE on an interval I provided there exists at least one function that satisfies the relation as well as the differential equation on I.

a relation or expression that defines a solution implicitly.

In contrast to an explicit solution


0),( =yxG

)(xy φ=

φ

φ

0),( =yxG


DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation


Cyxxxyy =−−−+ 232 22

0)22(34 ' =−++−− yyxxy


DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation


Cyxxxyy =−−−+ 232 22

0)22(34 ' =−++−− yyxxy

0)22(34

02234

02342

/)(/)232(

'

'''

'''

22

=−++−−==>=−++−−==>=−−−++==>

=−−−+

yyxxy

yyyxyxy

yxxyyyy

dxCddxyxxxyyd


Conditions Init ial Condition: Constrains that are specified at the initial point, generally time point, are called initial conditions. Problems with specified initial conditions

are called initial value problems.

Boundary Condition: Constrains that are specified at the boundary points, generally space points, are called boundary conditions. Problems with specified boundary conditions are called boundary value

problems.



First- and Second-Order IVPS Solve:

Subject to:

Solve:

Subject to:

1.2 Init ial-Value Problem

00 )( yxy =

),( yxfdx

dy =

),,( '2

2

yyxfdx

yd =

10'

00 )(,)( yxyyxy ==admission.edhole.com

DEFINITION: init ial value problem An init ial value problem or IVP is a

problem which consists of an n-th order ordinary differential equation along with n initial conditions defined at a point found in the interval of definition differential equation initial conditions

where are known constants.

1.2 Init ial-Value Problem

0x

),...,,,( )1(' −= nn

n

yyyxfdx

yd

I

10)1(

10'

00 )(,...,)(,)( −− === nn yxyyxyyxy

110 ,...,, −nyyyadmission.edhole.com

1.2 Init ial-Value ProblemTHEOREM: Existence of a Unique Solution

Let R be a rectangular region in the xy-plane defined by that containsthe point in its interior. If and are continuous on R, Then there

exists some interval contained in anda unique function defined on that is a solution of the initial value problem.

dycbxa ≤≤≤≤ ,

),( 00 yx ),( yxf

yf ∂∂ /

0,: 000 >+<<− hhxxhxI

bxa ≤≤)(xy

0I


b.tech admissions in delhi

Education