introduction to ode

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Differential Equations as Mathematical Models

Intr du ti n:

Mathematicalmodelsaremathematical

.

Level

of

resolutionMakesomereasonableassum tionsabout

thesystem.

Thestepsofmodelingprocessareasfollowing

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Steps in Constructing Mathematical Models

letters to represent them.

Choose the units of measure for each variable.

Articulate the basic principle that underlies or governs theproblem you are investigating. This requires your being

.

Express the principle or law in the previous step in terms ofthe variables identified at the start. This ma involve the useof intermediate variables related to the primary variables.

Make sure each term of your equation has the same physicalun ts.

The result may involve one or more differential equations.

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MathematicsEx ress assum tions in termsssump ons

formulationof different equations

If necessary,

alter assumptions Solve the DEs

of the model

Check model

predictions

Obtain

solutionDisplay model predictions,

e.g., graphically

facts

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Example 1: Free Fall

Formulate a differential equation describing motion of an

object falling in the atmosphere near sea level.

Variables: time t, velocity v

Newtons 2ndLaw: F= ma = m(dv/dt) net force

Force of gravity: F= mg downward force

Force of air resistance: F= v upward force Then

a ng g = . m sec , m = g, = g sec,

we obtain vv 2.08.9

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Example 1: Sketching Direction Field

Using differential equation and table, plot slopes (estimates)

on axes below. The resulting graph is called a direction

field. (Note that values of v do not depend on t.)v v'

0 9.8

5 8.8

.

15 6.8

20 5.8

.

30 3.8

35 2.8

40 1.8

45 0.8

50 -0.2

55 -1.2

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Example 1: Sketching Direction Field

Sample Matlab commands for graphing a direction field:

>>[x,y]=meshgrid(0:.5:10,0:5:80);

>>dy = 9.8 0.2*y;x=ones s ze y ;

>>quiver(x,y,dx,dy);

When graphing direction fields, be sure to use an appropriate

window, in order to display all equilibrium solutions and

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relevant solution behavior.


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Example 1: Direction Field & Equilibrium Solution

Arrows give tangent lines to solution curves, and indicate

where soln is increasing & decreasing (and by how much).

Horizontal solution curves are called equilibrium solutions. Use the graph below to solve for equilibrium solution, and

then determine analytically by setting v'= 0.

:0Set

v

8.9

..

v

49

.

v

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Example 2: Mice and Owls

Consider a mouse population that reproduces at a rate proportional

to the current population, with a rate constant equal to 0.5

mice/month (assuming no owls present). en ow s are presen , ey ea e m ce. uppose a e ow s

eat 15 per day (average). Write a differential equation describing

.

30 days in a month.)

Solution:

4505.0 p

dt

dp

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Example 3: Draining a Tank

FromTorricellisLaw,thespeedofthewater

leavingthetankis ,thenvolumeofgh2

.

V(t)denotesthevolumeofwaterinthetankh

,

hAdV

2

Since we haveV t =A h then

t

ghAdh h

2

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tw


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Example 4: A Slipping Chain

Weightofthechain W L

Massofthechain 232 32 /m W g L g ft s

LL

xxx22

We

ave2

or232 2

xdt

064

2

2

xxd

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Solution to General Equation

To solve the general equation

we use methods of calculus, as follows.

bayy

dtaaby

dya

aby

dtdy

a

bya

dt

dy

//

/

CatCat

Cat

ecceabeeab

eabyCtaaby

//

//ln

Thus the general solution is

,atcea

y

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Initial Value Problem

Next, we solve the initial value problem

0)0(, yybayy

atceaby

,

bycce

byy 0

0

0)0(

and hence the solution to the initial value problem is

aa

ate

a

by

a

by

0

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Equilibrium Solutions

,

'

,bayy

bty )(

Example: Find the equilibrium solutions of the following.

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Example 2: Solution of Initial Value Problem

We have infinitel man solutions to our e uation

since kis an arbitrar constant.

,9004505.0 5.0 tceppp

Graphs of solutions (integral curves) for several values of c,

and direction field for differential e uation, are iven below.

Choosing c = 0, we obtain the equilibrium solution, while forc 0, the solutions diverge from equilibrium solution.

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Example 2: Solution of Initial Value Problem

A differential equation often has infinitely many solutions. If

a point on the solution curve is known, such as an initial

condition, then this determines a unique solution. In the mice/owl differential equation, suppose we know that

the mice population starts out at 850. Thenp(0) = 850, and

t

cep

cetp0

5.0

900850)0(

900)(

c

:Solution

50

tetp5.050900)(

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Solutions to Differential Equations

A solution (t) to an ordinary differential equation

satisfies the equation:

)1()( ,,,,,)( nn yyyytfty

)1()( ,,,,,)( nn tft

ttyttyttyyy sin2)(,cos)(,sin)(;0 321

Three important questions in the study of differential

equations: Is there a solution? (Existence)

If there is a solution, is it unique? (Uniqueness)

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,

(Analytical Solution, Numerical Approximation, etc)

introduction to ode

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