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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 ;