chap 1 first-order differential equations
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王 俊 鑫 (Chun-Hsin Wang)
中華大學 資訊工程系
Fall 2002
Chap 1 First-Order Differential Equations
Chap 1 First-Order Differential Equations
Page 2
Outline
Basic Concepts
Separable Differential Equations substitution Methods
Exact Differential Equations Integrating Factors
Linear Differential Equations Bernoulli Equations
Page 3
Basic Concepts
Differentiation
x
ex
xx
aaa
ee
nxx
aa
xx
xx
nn
log)(log
1)(ln
ln)(
)(
)( 1
xxx
xxx
xx
xx
xx
xx
cotcsc)(csc
tansec)(sec
csc)(cot
sec)(tan
sin)(cos
cos)(sin
2
2
Page 4
Basic Concepts
Differentiation
xx
xx
sinh)(cosh
cosh)(sinh
21
21
2
1
2
1
1
1)(cot
1
1)(tan
1
1)(cos
1
1)(sin
xx
xx
xx
xx
Page 5
Basic Concepts
Integration
ca
adxa
cedxe
cxdxx
dxx
cn
xdxx
xx
xx
nn
ln
ln1
1
1
1
vdxuuvdxvu
vduuvudv
udxccudx
vdxudxdxvu )(
Page 6
Basic Concepts
Integration
cxxxdx
cxxxdx
cxxdx
cxxdx
cxxdx
cxxdx
cotcsclncsc
tanseclnsec
sinlncot
coslntan
sincos
cossin
Page 7
Basic Concepts
Integration
ca
xdx
ax
ca
xdx
ax
ca
xdx
xa
ca
x
adx
ax
1
22
1
22
1
22
122
cosh1
sinh1
sin1
tan11
Page 8
Basic Concepts
ODE vs. PDE
Dependent Variables vs. Independent
Variables
Order
Linear vs. Nonlinear
Solutions
Page 9
Basic Concepts
Ordinary Differential Equations An unknown function (dependent variable) y
of one independent variable x
xdx
dyy cos
04 yy
222 )2(2 yxyeyyx x
Page 10
Basic Concepts
Partial Differential Equations An unknown function (dependent variable)
z of two or more independent variables (e.g. x and y)
yxx
z46
yxyx
z
22
Page 11
Basic Concepts
The order of a differential equation is the order of the highest derivative that appears in the equation.
0)( 223 ynxyxyx Order 2
22
1
yxdx
dy
Order 1
1)( 432
2
ydx
ydOrder 2
Page 12
Basic Concept
The first-order differential equation contain only y’
and may contain y and given function of x.
A solution of a given first-order differential equation (*) on some open interval a<x<b is a function
y=h(x) that has a derivative y’=h(x) and satisfies (*) for all x in that interval.
),('
0)',,(
yxFy
yyxF
or (*)
Page 13
Basic Concept
Example : Verify the solution
x2
y
2yxy'
Page 14
Basic Concepts
Explicit Solution
Implicit Solution
)(xhy
0),( yxH
Page 15
Basic Concept
General solution vs. Particular solution
General solution arbitrary constant c
Particular solution choose a specific c
,....2,3
'
c
csinxy
cosxy
Page 16
Basic Concept
Singular solutions Def : A differential equation may sometimes have an
additional solution that cannot be obtained from the general solution and is then called a singular solution.
Example
The general solution : y=cx-c2
A singular solution : y=x2/4
0' yxyy' 2
Page 17
Basic Concepts
General Solution
Particular Solution for y(0)=2 (initial condition)
ktcety )(
ktety 2)(
kyy
Page 18
Basic Concept
Def: A differential equation together with an initial condition is called an initial value problem
00)(),,(' yxyyxfy
Page 19
Separable Differential Equations
Def: A first-order differential equation of
the form
is called a separable differential
equation
dxxfdyyg
f(x)g(y)y
)()(
'
Page 20
Separable Differential Equations
Example :
Sol:
049 xyy
Page 21
Separable Differential Equations
Example :
Sol:
21 yy
Page 22
Separable Differential Equations
Example :
Sol:
kyy
Page 23
Separable Differential Equations
Example :
Sol:
1)0(,2 yxyy
Page 24
Separable Differential Equations
Substitution Method:
A differential equation of the form
can be transformed into a separable
differential equation
)(x
ygy
Page 25
Separable Differential Equations
Substitution Method:
uxy uxuy
x
dx
uug
du
uugxu
uguxu
)(
)(
)(
Page 26
Separable Differential Equations
Example :
Sol:
222 xyyxy
cxyx
x
c
x
y
x
cu
cx
cxu
x
dx
u
uduu
uuxu
y
x
x
y
xy
x
xy
yy
xyyxy
22
2
2
112
2
22
22
1
1
1lnln)1ln(
1
2
)1
(2
1
)(2
1
22
2
Page 27
Separable Differential Equations
Exercise 1
201.01 yy
2/xyy
yyyx 2
2)2(,0' yyxy
Page 28
Exact Differential Equations
Def: A first-order differential equation of
the form
is said to be exact if
0),(),( dyyxNdxyxM
x
yxN
y
yxM
),(),(
Page 29
Exact Differential Equations
Proof:
0),(),(
0),(
dyyxNdxyxM
dyy
udx
x
uyxdu
x
yxN
y
yxM
yx
yxu
),(),(),(
Page 30
Exact Differential Equations
Example :
Sol:
0)3()3( 3223 dyyyxdxxyx
Exactxyx
N
y
M
xyx
yyx
xyy
xyx
,6
63
63
32
23
Page 31
Exact Differential Equations
Sol:
)(2
3
4
1
)()3(
)(
224
23
ykyxx
ykdxxyx
ykMdxu
1
4
322
4)(
3)(
3
cy
yk
yyxNdy
ydkyx
y
u
Page 32
Exact Differential Equations
Sol: cyyxxyxu )6(
4
1),( 4224
Page 33
Exact Differential Equations
Example
3)0(
0)sinh(cos)cosh(sin
y
dyyxdxyx
Page 34
Non-Exactness
Example : 0 xdyydx
Page 35
Integrating Factor
Def: A first-order differential equation of the form
is not exact, but it will be exact if multiplied by F(x, y)
then F(x,y) is called an integrating factor of this equation
0),(),( dyyxQdxyxP
0),(),(),(),( dyyxQyxFdxyxPyxF
Page 36
Exact Differential Equations
How to find integrating factor
Golden Rule
xxyy FQQFFPPF
Exactx
FQ
y
FP
FQdyFPdx
,
0
)(11
0
Let
xy
xy
QPQdx
dF
F
FQQdx
dFFPP
F(x)F
Page 37
Exact Differential Equations
Example :
Sol:
0 xdyydx
Exactx
N
xy
M
dyx
dxx
y
x
xdyydxx
F
,1
1
1
2
22
2
Page 38
Exact Differential Equations
Sol:
cxy
cx
yx
yddy
xdx
x
y
0)(1
2
Page 39
Exact Differential Equations
Example :
2)2(
0)cos()sin(2 22
y
dyyxydxy
Page 40
Exact Differential Equations
Exercise 2
02 2 dyxxydx 0)( 22 drrdre
xeFydyydx ,0cossinba yxFxdybydxa ,0)1()1(
0)1()1( dyxdxy
Page 41
Linear Differential Equations
Def: A first-order differential equation is
said to be linear if it can be written
If r(x) = 0, this equation is said to be
homogeneous
)()( xryxpy
Page 42
Linear Differential Equations
How to solve first-order linear homogeneous
ODE ?
Sol:
0)( yxpy
dxxpcdxxpcdxxp ceeeey
cdxxpy
dxxpy
dy
yxpdx
dy
)()()(
1
11
)(ln
)(
0)(
Page 43
Linear Differential Equations
Example :
Sol:
0 yy
x
cx
cx
dx
dxxp
ec
ece
ce
ce
cexy
2
)1(
)(
1
1
)(
Page 44
Linear Differential Equations
How to solve first-order linear nonhomogeneous
ODE ?
Sol:
)()( xryxpy
)())()(()(11
0))()((
)()(
xpxryxpy
QPQdx
dF
F
dydxxryxp
xryxpdx
dy
xy
Page 45
Linear Differential Equations
Sol:
dxxpexF
)()(
cdxreexy
cdxreye
reyepyye
dxxpdxxp
dxxpdxxp
dxxpdxxpdxxp
)()(
)()(
)()()(
)(
)()(
Page 46
Linear Differential Equations
Example :
Sol:
xeyy 2
xx
xx
xxx
xdxdx
dxxpdxxp
ece
cee
cdxeee
cdxeee
cdxreexy
2
2
2)1()1(
)()()(
Page 47
Linear Differential Equations
Example :
)2cos22sin3(2 xxeyy x'
Page 48
Bernoulli, Jocob
Bernoulli, Jocob1654-1705
Page 49
Linear Differential Equations
Def: Bernoulli equations
If a = 0, Bernoulli Eq. => First Order
Linear Eq.
If a <> 0, let u = y1-a
ayxgyxpy )()(
gapuau )1()1(
Page 50
Linear Differential Equations
Example :
Sol:
2ByAyy
AB
ceuy
A
Bcecdxe
A
BecdxBeeu
BAuu
AyBAyByyyyu
yyyu
Ax
AxAxAxAxAx
a
11
)( 1222
1211
Page 51
Linear Differential Equations
Exercise 3
4 yy kxekyy
22 yyy
1 xyxyy
)2(,sin3 yxyy
Page 52
Summary
可分離 Separable
變換法 Substitution
正合 Exact
積分因子 Integrating Factor
線性 Linear
柏努利 Bernoulli
dxxfdyyg )()(
dxxfduug )()(
0),(),( dyyxNdxyxM
0 FQdyFPdx
)()( xryxpy
ayxgyxpy )()(
Page 53
Orthogonal Trajectories of Curves
Angle of intersection of two curves is defined to be the angle between the tangents of the curves at the point of intersection
How to use differential equations for finding curves that intersect given curves at right angles ?
Page 54
How to find Orthogonal Trajectories
1st Step: find a differential equation for a given cure 2nd Step: the differential equation of the
orthogonal trajectories to be found
3rd step: solve the differential equation as above ( in 2nd step)
),( yxfy
),( yxfy'
),(
1
yxfy'
Page 55
Orthogonal Trajectories of Curves
Example: given a curve y=cx2, where c is arbitrary. Find their orthogonal trajectories.
Sol:
Page 56
Existance and Uniqueness of Solution
An initial value problem may have no solutions, precisely one solution, or more than one solution.
Example
1)0(,0' yyy
1)0(,' yxy
1)0(,1' yyxy
No solutions
Precisely one solutions
More than one solutions
Page 57
Existence and uniqueness theorems
Problem of existence Under what conditions does an initial
value problem have at least one solution ?
Existence theorem, see page 53 Problem of uniqueness
Under what conditions does that the problem have at most one solution ?
Uniqueness theorem, see page54
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