Chapter 1: First-Order Differential Equations

1

Sec 1.1: Differential Equations and Mathematical Models

Definition: Differential Equation

An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables, is said to be a differential equation (DE)

xxxxfy logsin)( 3 variableDependent

t variableIndependen

xxx

dxdyy 1cos3' 2

y' Find

:Example

22 txdtdx

1

0732

2

ydxdy

dxyd

2

2

Sec 1.1: Differential Equations and Mathematical Models

:Example )cos(''' 2 xyxy 2

dzduzu )ln(3

2

2

dxydexy x 4

)ln()ln())cot(csc())cos(5( xyyxxxex x 5

)ln()ln()5( swwsewdsd s 6

)ln())ln(5( swswesdsd s 7

equation Algebraic3

Sec 1.1: Definitions and Terminology

:Example

yu

xuu

8dxdw

dxudwu 3

329

Classification

Classification By

OrderType

4

Classification By Type

If an equation containing only ordinary derivatives it is said to be Ordinary Differential Equation (ODE)

An equation involving partial derivatives it is said to be Partial Differential Equation (PDE)

yu

xuu

8xeydxdy

51

Classification By Type (ODE,PDE)

5

Classification By OrderThe order of a differential equation (ODE or PDE) is the order of the highest derivative highest derivative in the equation. :Example

'''5' yeyy x1

2

245 2)(

dxydey

dxdy x2

3

3

2

24

yu

xuu

3

)5()( 22

24 uw

ywu

xu

4

0),....,'',',,( )( nyyyyxF5

Classification By Order

. n-th order DE6

highest derivativeODE or PDE

Classification By

OrderType

Classification

7

System of DE

)sin(''')cos('''

xyuxuy

System of two Ordinary Differential Equations2ed order, linear, ODE

8

Navier-Stokes Equations

ODE or PDE order?? ),(

Re1

),(Re1

22

22

122

2

22

2

11

21

121

2

21

2

yxfyp

yuu

xuu

yu

xu

yxfxp

yuu

xuu

yu

xu

One Million Dollar

9

Definition: Solution of an ODE

A continuous function is said to be a solution of a DE if it satisfies the DE on an interval I

)(xuu

:Remark

number real a is 012 equation algebraican ofsolution 2 xxfunction a is ODE ofsolution

:Example 0'2'' :ODE yyy xey(x) :sol

Sec 1.1: Differential Equations and Mathematical Models

10

:Example

0 allfor 04 :equation aldifferenti thesatisfies ln2 :function t theVerify tha 2121

xyxy'' x-xxy(x) //

:Example

xyy'Cey(x) x

2 :equation aldifferenti thesatisfies :function t theVerify tha

2

Families of Solutions

What diff?????

Sec 1.1: Differential Equations and Mathematical Models

Example of DE with no solution

11

:Example

xyy'Cey(x) x

2 :equation aldifferenti thesatisfies :function t theVerify tha

2

:Example

2 :equation aldifferenti the

satisfies )/(1 :function t theVerify tha

yy'

xCy(x)

What diff?????

Sec 1.1: Differential Equations and Mathematical Models

12

:Example

2 :equation aldifferenti the

satisfies )/(1 :function t theVerify tha

yy'

xCy(x)

:Example

09' :equation aldifferenti thesatisfies 3sin3cos :function t theVerify tha

yy'xBxAy(x)

What diff?????

Sec 1.1: Differential Equations and Mathematical Models

One-parameter family of solutions

Two-parameter family of solutions

13

14

Initial Value Problem

(IVP) Problem Value Initial Condition Initial

Equation alDifferenti

2)1(

2

yyy'

)/(1 xCy(x)

:Example Solve the IVP

-2 -1 0 1 2 3 4 5 6

-2

-1

0

1

2

x

2/(3-2 x)

15

The central question of greatest immediate interest to us is this : if we ate given a differential equation known to have a solution satisfying a given initial condition . How do we actually find or compute that solution? And, once found, what can we do with it? We will see that a relatively few simple technique

Separation of variables 1.4 Solution of linear equations 1.5 Elementary substitution method 1.6

Are enough to enable us to solve a varity of first-order equations having impressive applications.

Page 8

16

The study of DE has 3 principal goals

1) To discover the differential equation that describes a specified physical situation

2) To find either exactly or approximately the appropriate solution of that equation

3) To interpret the solution that is found

Find DE 1)0(

'

y

yy

Find sol

xexy )(

interpret sol

-4 -3 -2 -1 0 1 2 3 4 5

0

10

20

30

40

50

60

70

80

90

x

exp(x)

17

Differential Equations and Mathematical Models

:2Example4/p

Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank.

Scientific lawsScientific principals

Differential Equations

translate

18

Differential Equations and Mathematical Models

:2Example4/p Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank.

The The time rate of change of the volume V(t) of water in a draining

is proportional to the the square root of the depth y of water in the tank

dtdV y

ykdtdV

19

Differential Equations and Mathematical Models

Scientific lawsScientific principals

Differential Equations

translate

:2Example3/p Newton’s law of cooling: The time rate of change of the temperature T(t) of a body is proportional to the difference between T and the temperature A of surrounding medium

20

Differential Equations and Mathematical Models

The time rate of change of the temperature T(t) of a body

is proportional to the difference between T and the temperature A of surrounding medium

dtdT A)-(T

)(dtdT ATk

21

Differential Equations and Mathematical Models

:2Example4/p

Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank.

22

Differential Equations and Mathematical Models

:2Example4/p Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank.

The The time rate of change of the volume V(t) of water in a draining

is proportional to the the square root of the depth y of water in the tank

dtdV y

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