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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'