chap 1 first-order differential equations

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

Outline

Basic Concepts

Separable Differential Equations substitution Methods

Exact Differential Equations Integrating Factors

Linear Differential Equations Bernoulli Equations

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

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

Basic Concepts

Integration

ca

adxa

cedxe

cxdxx

dxx

cn

xdxx

xx

xx

nn

ln

ln1

1

1

1

vdxuuvdxvu

vduuvudv

udxccudx

vdxudxdxvu )(

Basic Concepts

Integration

cxxxdx

cxxxdx

cxxdx

cxxdx

cxxdx

cxxdx

cotcsclncsc

tanseclnsec

sinlncot

coslntan

sincos

cossin

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

Basic Concepts

ODE vs. PDE

Dependent Variables vs. Independent

Variables

Order

Linear vs. Nonlinear

Solutions

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

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

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

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 (*)

Basic Concept

Example : Verify the solution

x2

y

2yxy'

Basic Concepts

Explicit Solution

Implicit Solution

)(xhy

0),( yxH

Basic Concept

General solution vs. Particular solution

General solution arbitrary constant c

Particular solution choose a specific c

,....2,3

'

c

csinxy

cosxy

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

Basic Concepts

General Solution

Particular Solution for y(0)=2 (initial condition)

ktcety )(

ktety 2)(

kyy

Basic Concept

Def: A differential equation together with an initial condition is called an initial value problem

00)(),,(' yxyyxfy

Separable Differential Equations

Def: A first-order differential equation of

the form

is called a separable differential

equation

dxxfdyyg

f(x)g(y)y

)()(

'


Example :

Sol:

049 xyy


Example :

Sol:

21 yy


Example :

Sol:

kyy


Example :

Sol:

1)0(,2 yxyy


Substitution Method:

A differential equation of the form

can be transformed into a separable

differential equation

)(x

ygy


Substitution Method:

uxy uxuy

x

dx

uug

du

uugxu

uguxu

)(

)(

)(


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


Exercise 1

201.01 yy

2/xyy

yyyx 2

2)2(,0' yyxy

Exact Differential Equations

Def: A first-order differential equation of

the form

is said to be exact if

0),(),( dyyxNdxyxM

x

yxN

y

yxM

),(),(


Proof:

0),(),(

0),(

dyyxNdxyxM

dyy

udx

x

uyxdu

x

yxN

y

yxM

yx

yxu

),(),(),(


Example :

Sol:

0)3()3( 3223 dyyyxdxxyx

Exactxyx

N

y

M

xyx

yyx

xyy

xyx

,6

63

63

32

23


Sol:

)(2

3

4

1

)()3(

)(

224

23

ykyxx

ykdxxyx

ykMdxu

1

4

322

4)(

3)(

3

cy

yk

yyxNdy

ydkyx

y

u


Sol: cyyxxyxu )6(

4

1),( 4224


Example

3)0(

0)sinh(cos)cosh(sin

y

dyyxdxyx

Non-Exactness

Example : 0 xdyydx

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


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


Example :

Sol:

0 xdyydx

Exactx

N

xy

M

dyx

dxx

y

x

xdyydxx

F

,1

1

1

2

22

2


Sol:

cxy

cx

yx

yddy

xdx

x

y

0)(1

2


Example :

2)2(

0)cos()sin(2 22

y

dyyxydxy


Exercise 2

02 2 dyxxydx 0)( 22 drrdre

xeFydyydx ,0cossinba yxFxdybydxa ,0)1()1(

0)1()1( dyxdxy

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


How to solve first-order linear homogeneous

ODE ?

Sol:

0)( yxpy

dxxpcdxxpcdxxp ceeeey

cdxxpy

dxxpy

dy

yxpdx

dy

)()()(

1

11

)(ln

)(

0)(


Example :

Sol:

0 yy

x

cx

cx

dx

dxxp

ec

ece

ce

ce

cexy

2

)1(

)(

1

1

)(


How to solve first-order linear nonhomogeneous

ODE ?

Sol:

)()( xryxpy

)())()(()(11

0))()((

)()(

xpxryxpy

QPQdx

dF

F

dydxxryxp

xryxpdx

dy

xy


Sol:

dxxpexF

)()(

cdxreexy

cdxreye

reyepyye

dxxpdxxp

dxxpdxxp

dxxpdxxpdxxp

)()(

)()(

)()()(

)(

)()(


Example :

Sol:

xeyy 2

xx

xx

xxx

xdxdx

dxxpdxxp

ece

cee

cdxeee

cdxeee

cdxreexy

2

2

2)1()1(

)()()(


Example :

)2cos22sin3(2 xxeyy x'

Bernoulli, Jocob

Bernoulli, Jocob1654-1705


Def: Bernoulli equations

If a = 0, Bernoulli Eq. => First Order

Linear Eq.

If a <> 0, let u = y1-a

ayxgyxpy )()(

gapuau )1()1(


Example :

Sol:

2ByAyy

AB

ceuy

A

Bcecdxe

A

BecdxBeeu

BAuu

AyBAyByyyyu

yyyu

Ax

AxAxAxAxAx

a

11

)( 1222

1211


Exercise 3

4 yy kxekyy

22 yyy

1 xyxyy

)2(,sin3 yxyy

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 )()(

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 ?

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'

Orthogonal Trajectories of Curves

Example: given a curve y=cx2, where c is arbitrary. Find their orthogonal trajectories.

Sol:

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

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

chap 1 first-order differential equations

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