l1 first order odes

7/21/2019 L1 First Order ODEs

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Dr. Tanveer Iqbal

First Order ODEs


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Mathematical Modeling


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Diferential Equations

Ordinary Partial

An equation that contains one orseveral derivatives o an unknon

unction

!DEs involve "artial derivatives o anunknon unction o two or more

variables


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&olution 'urves

Initial (alue !roblem


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.Decay

Given an amount of a radioactive

substance, say, 0.5 g (gram), nd theamount resent at any later time. !hysical"nformation. #xeriments show that at eachinstant a radioactive substance

decomoses and is thus decaying in timeroortional to the amount of substanceresent.

Step 1. Setting up a mathematical

model of the physical process.

Now the given initial amount is 0.5 g,and we can call the correspondinginstant t=0


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Radioactivity. ExponentialDecay

Step 2. athematical solution

!lways check your result

Step ".

#nterpretation of result.


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Direction FieldsDerivative is slo"e o curve


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)umerical Method*Euler$sMethod+iven an ODEAn initial value

Eulers method yields approximate

solution values at equidistantx$values, h,as


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umer ca e o * u er sMethod


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umer ca e o * u er sMethod

,et ODE ith initial condition #-/ 0(eri# the solution is


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&e"arable ODEs1Method o &e"arating(ariable


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Method o &e"arating(ariable


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Method o &e"arating (ariable*Modeling 2adioactive Deca#

In &e"tember 3443 the amous Iceman -Oet5i/6 a mumm#rom the )eolithic "eriod o the &tone Age ound in the ice othe Oet5tal Al"s -hence the name 7Oet5i8/ in &outhern T#rolianear the Austrian9Italian border6 caused a scienti:c sensation.;hen did Oet5i a""ro%imatel# live and die i the ratio o

carbon3.?@ o that o aliving organism

$hysical #nformation. "n the atmoshere and in livingorganisms, the ratio of radioactive carbon 3


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2adioactive Deca#

odeling. &adioactive decay is governed by the1#

# se"aration and integration -here t is time

and y0is the initial ratio of3 A.M.

Step 5. !nswer and interpretation

7 8.9. is t3 (namely, hours after 0 !.9.)

Bence the tem"erature in the building dro""ed 4F


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Method o &e"arating (ariable*Modeling ,eaking Tank

The "roblem is concerned theouto o ater rom ac#lindrical tank ith a hole atthe bottom. Jou are asked to

:nd the height o the aterin the tank at an# time i thetank has diameter > m6 thehole has diameter 3 cm6 and

the initial height o the aterhen the hole is o"ened is>.>? m. ;hen ill the tankbe em"t#


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Modeling ,eaking Tank!hysical information. Hnder the inuence o

gravit# the outoing ater has velocit#($orricellis la!#%

;here6 h is the height o the ater above thehole at time t, and is the acceleration o

gravit# at the surace o the earth.Step 1. Setting up the model.2elate the decrease in ater level h to the

outo. The volume K( o the outo during a

short time Kt is


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Modeling ,eaking TankStep 2. %eneral Solution

Step " $articular Solution -Find c romInitial 'ondition/

&tep ' Empty $an


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E%tended MethodL 2eduction to&e"arable Form

,et the equation be


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E t ODE


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E%act ODEs

A :rst order diferential equation

Is e%act. I

&olution or E%act ODE

Bere6 k-#/ is an integration constant.

To :nd k :nd "artial derivative o u ithres"ect # and set it equal to ).

E ODE E l


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E%act ODE* E%am"leSolve

Step 1) *est for +actness

Step 2. #mplicit general solution

'etermine

Step ". (hecking an implicit solution.

Factor


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Factor

I equation

is non*e%act than

'an be e%act here F an integrating actor6

and unction o % and #.E%am"le

I t ti F t


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Integrating Factors

ased on condition o e%actness

Than6

Theorem 3. I F is unction o % onl#

I not than take F instead o F and dF1d# ordetermining 2. Than use

Theorem >. I F is unction o # onl#

F t


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Factor

Step 2. #ntegrating factor. %eneral

solution.


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,inear ODEsA :rst*order ODE is said to be linear i it can be

brought into the orm b# algebra6

)onlinear i it cannot be brought into this orm

I linear equation becomes

Than it is a homogeneous linear equation.

Bomogeneous ,inear Equations can be solvedb# se"arating variables method.

ODE


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ODEs


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Electric 'ircuit

9odel the &6$circuit in :ig. below and solvethe resulting 1# for the current "(t) 8(ameres), where t is time. 8ssume thatthe circuit contains as an #9:

(electromotive force) a battery of #3 ;

l1 first order odes

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