第一章 一階常微分方程式 part 1 -...
Transcript of 第一章 一階常微分方程式 part 1 -...
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: part 1
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://modeling
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()Differential Equation :
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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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()