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Engineering Mathematics Ⅰ

呂學育 博士Oct. 13, 2004

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1.5 Integrating Factors • The equationis not exact on any rectangle.Because and

and

=The equation is not exact on any rectangle.

0)63(6 '22 yxxyxyy

xyyM 62 263 xxyN

xyxyyyy

M 62)6( 2

xyxxy

xxN 123)63( 2

xN

yM

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1.5 Integrating Factor • Recall• Theorem 1.1 Test for Exactness is exact on if and only if (), for each in ,

0),(),( ' yyxNyxM R

),( yx R xN

yM

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1.5 Integrating Factor If is exact, then there is a potential function and

and (*)

implicitly defines a function y(x) that is general solution of the differential equation.

Thus, finding a function that satisfies equation(*) is equivalent to solving the differential equation.

0),(),( ' yyxNyxMφ

),(φ yxMx

),(φ yxN

y

Cxyx )(,φ

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1.5 Integrating Factor • Definition 1.5

Let and be defined on a region of a plane. Then is an integrating factor for if for all in , and is exact on .

),( yxM ),( yxN R),(μ yx

0' yNM 0),(μ yx ),( yxR R0μμ ' yNM

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1.5 Integrating Factor • Example 1.21 is not exact.

Here and

For to be an integrating factor,

0' yxyx

xyxM 1N

xxyxyy

M

)( 0)1(

xxN

μ )μ()μ( My

Nx

)(μ)μ( xyxyx

μμ)(μ xy

xyxx

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1.5 Integrating Factor • Example 1.21For to be an integrating factor,

To simplify the equation, we try to find as

just a function of This is a separable equation.

μ

μμ)(μ xy

xyxx

x

?),(μ),(μ,)(μ yxyx

0μ

y

μ

μμ xx

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1.2 Separable Equations• A differential equation is called separable

if it can be written as• Such that we can separate the variables

and write

• We attempt to integrate this equation

)()(' yBxAy

dxxAdyyB

)()(

1 0)( yB

dxxAdyyB

)()(

1

Recall

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1.5 Integrating Factor • Example 1.21

Integrate to obtain to get one integrating factor

(*)

The equation (*) is exact over the entire plane, FOR ALL (x,y) !

μμ xx

xdxd μ

μ1

xdxd μμ1

Cx 2

21μln

02/2)(μ xex

0)( '2/2/ 22 yeexyx xx

2/2/ 22)( xx xeexyxyy

M

2/2/ 22 )( xx xee

xxN

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1.5 Integrating Factor If is exact, then there is a potential function and

and (*)

implicitly defines a function y(x) that is general solution of the differential equation.

Thus, finding a function that satisfies equation(*) is equivalent to solving the differential equation.

0),(),( ' yyxNyxMφ

),(φ yxMx

),(φ yxN

y

Cxyx )(,φ

Recall

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1.5 Integrating Factor • Example 1.21

and

Then we must have

The general solution of the original equation is

0)( '2/2/ 22 yeexyx xx

2/2)(),(φ xexyxyxMx

2/2),(φ xeyxN

y

)(),(φ 2/2/ 22 xhyedyeyx xx 2/2)(φ xexyx

x

)()( '2/2/ 22 xhxyexhye

xxx

2/' 2)( xxexh 2/2)( xexh 2/2)1(),(φ xeyyx

Ceyyx x 2/2)1(),(φ 2/21)( xCexy

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1.5.1 Separable Equations and Integrating Factor • The separable equationis in general not exact. Write it as and

In generalHowever, is an integrating factor for the separable equation. If we multiply the DE by ,we get

an exact equation. Because

)()(' yBxAy

0)()( ' yyBxA

)()()()( ' yBxAyBxAyy

M

0)1(

xxN

0)()( ' yBxA

)(1)(μ yBy

)(1)(μ yBy

0)(

1)( ' yyB

xA

0)()(

1

xA

yyBx

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1.3 Linear Differential Equations• Linear: A differential equation is called linear if

there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables.

• Non-linear: Differential equations that do not satisfy the definition of linear are non-linear.

• Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.

)()()(' xqyxpxy

Recall

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1.3 Linear Differential Equations• Example 1.14 is a linear DE. P(x)=1 and

q(x)=sin(x), both continuous for all x.An integrating factor is

Multiply the DE by to getOr

Integrate to get

The general solution is

)sin(' xyy

xdxdxxp eee )(

xe )sin(' xeyeey xxx )sin(' xeye xx

Cxxedxxeye xxx )cos()sin(21)sin(

xCexxy )cos()sin(21

Recall

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1.5.2 Linear Equations and Integrating Factor • The linear equationWrite it as andso the linear equation is not exact unless

However,

is exact because

)()(' xqyxpy

0)()( ' yxqyxp

)()()( xpxqyxpyy

M

01

xxN

0)( xpdxxpeyx )(),(μ

0)()( ')()( yeexqyxp dxxpdxxp

dxxpdxxpdxxp exqyxpy

expex

)()()( )]()([)(

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1.6 Homogeneous, Bernoulli, and Riccati Equations

1.6.1 Homogeneous Differential Equations

Definition 1.6.1 Homogeneous EquationA first-order differential equation is homogeneous if it

has the form

For example: is homogeneous

while is not.

xyfy '

xy

xyy sin'

yxy 2'

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1.6 Homogeneous, Bernoulli, and Riccati Equations

1.6.1 Homogeneous Differential EquationsA homogeneous equation is always transformed into a

separable one by the transformation

and write

Then becomes

And the variables (x, u) have been separated.

uxy

uxuuxxuy '''' xyu /

)/(' xyfy )(' ufuxu

xdxdu

uuf1

)(1

dxx

duuuf

1)(1

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1.6 Homogeneous, Bernoulli, and Riccati Equations

1.6.1 Homogeneous Differential EquationsExample 1.25

Let y=ux or

the general solution of the transformed equation

the general solution of the original equation

yxyxy 2

'

xy

xyy

2

'

uuuxu 2' 2' uxu

dxx

duu

112

Cxu

ln1

Cxxu

ln

1)(

Cxxy

ln

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1.6 Homogeneous, Bernoulli, and Riccati Equations

1.6.2 The Bernoulli EquationsA Bernoulli equation is a first-order equation (*) in which is a real

number.

If or separable and linear ODE

Let for , then (**)

(*)

(*),(**) linear ODE

α' )()( yxRyxPy α

0α

α1yv 1αdxdyy

dxdv α)α1(

α1α )()( yxPxRdxdyy )()(α xvPxR

dxdyy

)]()()[α1( xvPxRdxdv

1α

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1.6 Homogeneous, Bernoulli, and Riccati Equations1.6.2 The Bernoulli EquationsExample 1.27

which is Bernoulli with , and

Make the change of variables , then

and

so the DE becomes

upon multiplying by a linear equation

32' 31 yxyx

y

xxP /1)( 23)( xxR 3αα' )()( yxRyxPy

2yv 2/1vy

)(21)( '2/3' xvvxy

2/322/1'2/3 31)(21 vxv

xxvv

2/32v2' 62)( xv

xxv

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1.6 Homogeneous, Bernoulli, and Riccati Equations1.6.2 The Bernoulli EquationsExample 1.27

a linear equation

An integrating factor is

Integrate to get

The general sol of the Bernoulli equation is

32' 31 yxyx

y

2' 62)( xvx

xv

2)ln(2)/2()(),(μ xeeeyx xdxxdxxp

62)( 3'2 vxxvx 6'2 vx

Cxvx 62 236 Cxxv

32 61

)(1)(

xCxxvxy

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1.6 Homogeneous, Bernoulli, and Riccati Equations1.6.2 The Riccati EquationsDefinition 1.8 A differential equation of the form

is called a Riccati equationA Riccati equation is linearly exactly when

Consider the first-order DEIf we approximate ,while x is kept constant,

How to transform the Riccati equation to a linear one ?

)()()( 2' xRyxQyxPy

0)( xP

),( yxfdxdy

),( yxf

...)()()(),( 2 yxRyxQxPyxf

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1.6 Homogeneous, Bernoulli, and Riccati Equations1.6.2 The Riccati Equations

How to transform the Riccati equation to a linear one ?

Somehow we get one solution, , of a Riccati equation, then the change of variables

transforms the Riccati equation to a linear one.

)()()( 2' xRyxQyxPy

)(xS

zxSy 1)(

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1.6 Homogeneous, Bernoulli, and Riccati Equations1.6.2 The Riccati EquationsExample 1.28

By inspection, , is one solution. Define a new variable z by the change of variables

Then

Or a linear equation

xyx

yx

y 211 2'

1)( xSy

zy 11

'2

' 1 zz

y

xzxzxz

z21111111 2

'2

xzx

z 13'