: part 1

:

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://modeling

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()Differential Equation :

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()

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()

(Modeling)

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()

Differential Equationdependent variableindependent variable

(dep indep)

4.10

3.1

2.1sin35

1.10

2

2

2

2

2

2

2

2

4

4

2

2

2

=

+

+

=

+

=++

=

+

zu

yu

xu

vtv

sv

txdt

xddt

xddxdyxy

dxyd

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()

Ordinary Differential Equation

1.11.2 O.D.E.(1.1xy1.2tx)

2.1sin35

1.10

2

2

4

4

2

2

2

txdt

xddt

xddxdyxy

dxyd

=++

=

+

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()

Partial Differential Equation

1.31.4 P.D.E.(1.3stv

1.4x,y,zu)

4.10

3.1

2

2

2

2

2

2

=

+

+

=

+

zu

yu

xu

vtv

sv

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()

Differential EquationOrder

1.121.241.311.4 2

4.10

3.1

2.1sin35

1.10

2

2

2

2

2

2

2

2

4

4

2

2

2

=

+

+

=

+

=++

=

+

zu

yu

xu

vtv

sv

txdt

xddt

xddxdyxy

dxyd

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Linear Ordinary Differential

Equationn

ydegree1y

1.51.6

()

),()()()()()( 122

21

1

10 xbyxadxdyxa

dxydxa

dxydxa

dxydxa nnn

n

n

n

n

n

=+++++

L

6.1

5.1065

1.10

33

32

4

4

2

2

2

2

2

xxedxdyx

dxydx

dxyd

ydxdy

dxyd

dxdyxy

dxyd

=++

=++

=

+

1.1degree21.51.6

2

dxdy

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Nonlinear Ordinary Differential Equation

()

9.1065

8.1065

7.1065

2

2

3

2

2

22

2

=++

=+

+

=++

ydxdyy

dxyd

ydxdy

dxyd

ydxdy

dxyd

3

5

dxdy

dxdyy5

1.7 6y2 degree2 1.8 degree3 1.8 y

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()

0cos3sin2cos3sin2)(=

++==cxxy

cxxxfy

022

=+ ydx

yd

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()

Explicit Solution of the Ordinary Differential Equation

nFreal functionn+2 fx= y Ixnf1.10I

A. Ix

B. Ix

,10.10,,, =

n

n

dxyd

dxdyyxF L

[ ])(,),(),(, xfxfxfxF nL

[ ] 0)(,),(),(, = xfxfxfxF nL

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()

Explicit Solution of theOrdinary Differential Equation

x

xf(x)xf(x)

y f(x)1.12xf(x)1.12

0

11.1cos3sin2)( xxxf +=

12.1022

=+ ydx

yd

.cos3sin2)(,sin3cos2)(xxxf

xxxf==

2

2

dxyd )(xf

0)cos3sin2()cos3sin2( =++ xxxx

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()

Implicit Solution of the Ordinary Differential Equation

H(x,y)=0 y= f(x)1.10I H(x,y)=0 1.10

,10.10,,, =

n

n

dxyd

dxdyyxF L

0)()(25

)(

25)(,25)(

550025:

1121

22

21

22

=+

=

==

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()

0,0)()(25

)(

25)(,25)(

0025:

1121

22

21

22

=+=+

=

==

=+=++

dxdy

dxdy

yxxfxfxx

xxf

xxfxxf

yxyx

yxyx

yxxxfxxf

dxdy 0,025,

,,,25)(25)(

22

22

21

=+=++

==

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()

General Solution Particular Solution Singular Solution

O.D.E.

n O.D.E. n O.D.E.

O.D.E.

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()

General Solution Particular Solution Singular Solution

( )

( )

( )( ) ( )( )

xyx

xyy

cxcxcx

cxdxd

cxx

dxd

dxdy

cxxy

xydxdy

=

==

++=++=+=

+=

+=

+=

+=

:

1,0)1(,:

111111

111:

1:

222

1

2

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(Separable equation)

)2()()(

),1(

)1()()(

LL

LLL

dxxfdyyg

dxdy

y

xfyyg

=

=

)3()()(

)()()1(

LLLcdxxfdyyg

cdxxfdxdxdyygx

+=

+=

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()

18

2222

29 ,

492

49

049

cccyxcxy

xdxydy

xyy

==++=

=

=+

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()

0)1()1(x 22 =+++ yy

011

0)1()1(x0)1()1(x 222222 =

++

+=+++=+++

xdx

ydydxydyy

dxdy

0tantan011

1122 =++=++

+ cxyx

dxy

dy

cxy ,)tantan(tantan 11 =+

cyxxyc

xyxy

1,

)tan(tan)tan(tan1)tan(tan)tan(tan

11

11

=+

=

+

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()

2)1( ,0

==

yydx-xdy

00 ==y

dyxdxxdyydx

cyxccyx

ydy

xdx

yx lnlnln0 ====

212)1(: == cy

xyyx 2

21

==

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()

0)0(

2sin)x(os 2

=

=

y

xedxdyyec yy

xdxdyyee

xdxedyyexxedxdyyec

yy

yyyy

sin2)(

sin2)(cossin2)x(os 22

=

==

cxeye yy +=++ cos2 )1(

40)0(: == cy

4cos2 )1( +=++ xeye yy

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()

xdx

uugduuugxuuguxu

xygyuxuyuxy

xyu

xygy

=

==+

=+===

=

)()()(

,,

:,:

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()

xyulet =

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()

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()