engineering mathematics Ⅰ

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Engineering Mathematics Ⅰ

呂學育博士Oct. 13, 2004

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1.5 Integrating Factors

• The equation

is 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

M62)6( 2

xyxxyxx

N123)63( 2

x

N

y

M

3

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 x

N

y

M

4

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),(),( ' yyxNyxM

φ

),(φ

yxMx

),(φ

yxNy

Cxyx )(,φ

5

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 ),( yx

R R0μμ ' yNM

6

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(

xx

N

μ )μ()μ( My

Nx

)(μ)μ( xyxyx

μμ

)(μ

xy

xyxx

7


For 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

8

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

9


Integrate to obtain to get one integrating factor

(*)

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

μμ

xx

xdxd μ

μ

1xdxd μ

μ

1

Cx 2

2

1μln

02/2)(μ xex

0)( '2/2/ 22 yeexyx xx

2/2/ 22

)( xx xeexyxyy

M

2/2/ 22

)( xx xeexx

N

10

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),(),( ' yyxNyxM

φ

),(φ

yxMx

),(φ

yxNy

Cxyx )(,φ

Recall

11


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 xeyxNy

)(),(φ 2/2/ 22

xhyedyeyx xx 2/2)(

φ xexyxx

)()( '2/2/ 22

xhxyexhyex

xx

2/' 2

)( xxexh 2/2)( xexh 2/2)1(),(φ xeyyx

Ceyyx x 2/2)1(),(φ 2/21)( xCexy

12

1.5.1 Separable Equations and Integrating Factor • The separable equation

is 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(

xx

N

0)()( ' yBxA

)(1)(μ yBy

)(1)(μ yBy

0)(

1)( ' y

yBxA

0)()(

1

xAyyBx

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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

14

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 get

Or

Integrate to get

The general solution is

)sin(' xyy

xdxdxxp eee )(

xe )sin(' xeyeey xxx )sin(

'xeye xx

Cxxedxxeyexxx )cos()sin(

2

1)sin(

xCexxy )cos()sin(2

1

Recall

15

1.5.2 Linear Equations and Integrating Factor • The linear equation

Write it as

and

so the linear equation is not exact unless

However,

is exact because

)()(' xqyxpy

0)()( ' yxqyxp

)()()( xpxqyxpyy

M

01

xx

N

0)( xp

dxxpeyx )(),(μ

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

dxxpdxxpdxxp exqyxpy

expex

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

16

1.6 Homogeneous, Bernoulli, and Riccati Equations

1.6.1 Homogeneous Differential Equations

Definition 1.6.1 Homogeneous Equation

A first-order differential equation is homogeneous if it has the form

For example: is homogeneous

while is not.

x

yfy '

x

y

x

yy sin'

yxy 2'

17



A 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

xdx

du

uuf

1

)(

1

dxx

duuuf

1

)(

1

18



Example 1.25

Let y=ux or

the general solution of the transformed equation

the general solution of the original equation

yx

yxy

2'

x

y

x

yy

2

'

uuuxu 2' 2' uxu

dxx

duu

112

Cxu

ln1

Cxxu

ln

1)(

Cx

xy

ln

19


1.6.2 The Bernoulli Equations

A 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 1αdx

dyy

dx

dv α)α1(

α1α )()( yxPxRdx

dyy )()(α xvPxR

dx

dyy

)]()()[α1( xvPxRdx

dv

1α

20



Example 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

)(2

1)( '2/3' xvvxy

2/322/1'2/3 31

)(2

1 vxvx

xvv

2/32v2' 6

2)( xvx

xv

21



Example 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 6

1

)(

1)(

xCxxvxy

22


1.6.2 The Riccati Equations

Definition 1.8 A differential equation of the form

is called a Riccati equation

A Riccati equation is linearly exactly when

Consider the first-order DE

If we approximate ,while x is kept constant,

How to transform the Riccati equation to a linear one ?

)()()( 2' xRyxQyxPy

0)( xP

),( yxfdx

dy

),( yxf

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

23



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)(

24



Example 1.28

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

Then

Or

a linear equation

xyx

yx

y211 2'

1)( xSy

zy

11

'

2

' 1z

zy

xzxzxz

z

211

111

112

'

2

xzx

z13'

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