1-1st order ODE

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Basic Concepts and ClassifyingDierential Equations

TK-124 Matematika Teknik Kimia I

First Order Dierential Equations


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|||| 9.1 Modeling wit Dierential Equations

Perhaps the most important of all the applications ofcalculus

is to dierential equations.

When phsical scientists or social scientists usecalculus! more often than not it is to anal"e a

dierential equation that has arisen in the process ofmodelin# some phenomenon that the are studin#.

$lthou#h it is often impossi%le to &nd an e'plicitformula for the solution of a dierential equation! (e

(ill see that #raphical and numerical approaches


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Dierential Equations

$ diferential equation is an equation involving anunknown unction andits deri)ati)es.$ dierential

equation is anordinary diferentialequation i theunknownfunction depends on

onl oneindependent)aria%le. If theunkno(n function

depends on t(o ormore inde endent


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Order ! "olution

The order o a diferential equation is the order o thehi#hest deri)ati)e appearin# in the equation.

$ solution o a diferential equation in the unknownunction y and the

independent )aria%le x on the interval ℑ is a unction y(x) that satisesthe dierential equation identicall for all x in ℑ .

*ome dierential equations ha)e in&nitel mansolutions! (hereas other dierential equations ha)e nosolutions. It is also possi%le that a dierential equationhas e'actl one solution.

The general solution o a diferential equation is the set


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#nitial$%alue &s Boundary$%alue 'ro(le)s

$ dierential equation alon# (ith su%sidiar conditionson the unkno(n function and its deri)ati)es! all #i)enat the same )alue of the independent )aria%le!constitutes an initial-value problem. he subsidiaryconditions are initial conditions.

! the subsidiary conditions are given at more than one)alue of the independent )aria%le! the pro%lem is a

boundary-value problemand the conditions are boundary conditions.


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"tandard and DierentialFor)s

*tandard form for a &rst-order dierential equation inthe unkno(n function y(x) is

iferential orm


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First$Order Dierential

Equations


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"epara(le

Equations

+onsider a dierential equation in dierential form,

If #(x$y) % &(x) a function onl of x ) and '(x$y) % (y) (a unction only o y )$ the diferential equation isseparable$ or has its variables separated.

The #eneral solution to the &rst-order separa%ledierential equation,

is

re c represents an arbitrary constant.


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"epara(le Equations*E+a)ples

1. *ol)e

2. *ol)e

/. *ol)e

4. *ol)e


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E+ercise


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E+ercise


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"epara(le Equations*#llustration of ,pplications

ro(th and eca Pro%lemset '(t) denote the amount o substance (or

population) that is either #ro(in# or decain#. If (eassume that d' dt $ the time rate o change o this

amount o su%stance! is proportional to the amount ofsu%stance present! then d'dt % k'$ or

(here k is the constant o proportionality.

We are assumin# that '(t) is a diferentiable$ hencecontinuous$ unction of time. 3or population pro%lems!(here '(t) is actually discrete and inte#er-)alued! thisassumption is incorrect. onetheless!q. /.15 still

pro)ides a #ood appro'imation to the phsical la(s


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$ %acteria culture is kno(n to #ro( at a rateproportionalto the amount present. $fter one hour! 1666 strands of

the %acteria are o%ser)ed in the culture7 and after fourhours! /666 strands.3ind,(a)an expression or the approximate number o

strands o the bacteria present in the culture at antime t and

(b)the approximate number o strands of the %acteriaori#inall in the culture.$ssumin# that the #ro(th rate is proportional to

population si"e! de)elop model for the population ofthe (orld. In 1896! the population is 29:6 millions ;in 2666 increase to :6


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'opulation -rowt


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'opulation -rowt


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adioacti&e Decay


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More on 'opulation -rowt* /ogistic Model


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/ogistic DierentialEquation


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E+ercise


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E+ercise


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E+ercise


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E+ercise


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emperature Pro%lemse(ton>s la( of coolin#! (hich is equall applica%le toheatin#! statesthat the time rate o change o the temperature o a body

is proportionalto the temperature diference between the body and itssurrounding medium.

et denote the temperature o the body and let m

denote the temperature of the surroundin# medium. Then the time rate of chan#e of the temperature of the%od is d dt $ and 'ewton*s law o cooling can beormulated as d dt % +k( + m )$ or asre k is a positive constant o proportionality.


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0ewtons /aw of Cooling


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Mi'in# Pro%lem


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E+ercise


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+hemical ?eaction Kinetics


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eaction 2inetics


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eaction 2inetics


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eaction 2inetics

Pen#uraian hidro#en peroksida den#ankatalis adalah reaksi orde satu. Tetapanla@u pen#uraian hidro#en peroksida adalah

:!2 ' 16-4 s-1

a. Aitun# %erapa B hidro#en peroksida an#terurai dalam (aktu 26 menit

%. Aitun# (aktu paro tC! aitu (aktu an#diperlukan a#ar hidro#en peroksidaterurai se%anak 96B


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/D

eaction 2inetics

*uatu #as $ terurai menurut reaksi orde dua. Eika konsentrasi #as $ mula-mula 2!1 M dandalam (aktu 9 menit konsentrasina turun

men@adi 1!99 M!a. Aitun# har#a k%. Aitun# tCc. Perkirakan (aktu an# diperlukan a#ar

konsentrasi $ turun hin##a hana 6!1 M


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E+ercise


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rtho#onal Tra@ectories


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Ortogonal 3ra4ectories


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Ortogonal 3ra4ectories


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5o)ogeneous

Equations

ierential equation in standard form

homogeneous i

or e)er real num%er t.

In the #eneral frame(ork of dierential equations! the(ord Ghomo#eneousH has an entirel dierentmeanin#. Fnl in the conte't of &rst-order dierentialequations does Ghomo#eneousH ha)e the meanin#de&ned a%o)e.

5o)ogeneous


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5o)ogeneousEquations

can %e transformed into a separa%le equation %

makin# the su%stitution

on# (ith its correspondin# deri)ati)e


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E+a)ples

*ol)e This dierential equation is notsepara%le. Instead it is homo#eneous,

*u%stitutin# quations 2.: and 2.D into

the equation results in a separa%leequation


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E+act Equations

ierential equation in dierential form

is e'act if

thod of *olution, &rst sol)e the equationsand

for g(x$ y). he solution is then given implicitly by

*teps in sol)in# e'act dierential equations,


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1. Test for e'actness2. Pro)ide an implicit #eneral solution/. +heck the implicit solution

Equations


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Equations

in #eneral! is not e'act. Fccasionall! it is possi%le totransform 1.D5into an e'act dierential equation % a @udiciousmultiplication.$ function !(x$ y) is an integrating actor or (,.) i theequationis e'act. *ome of the more common inte#ratin# factorsare


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E+ercise


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E+ercise


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E+ercise


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E+ercise


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/inear Equations

ierential equation in standard form

is linear if (x$y)% +p(x)yq(x). 3irst-order lineardierential equations can al(as %e e'pressed as

inte#ratin# factor for quation 1.< is

(hich depends onl on x and is independent o y ./hen both sides o ,.0 are multi lied % ! x the


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/inear Equations/hen both sides o ,.0 are multiplied % !(x)$ theresulting equation

is e'act. This equation can %e sol)ed % the methoddescri%ed pre)iousl. $ simpler procedure is to re(rite2.2/ as

inte#rate %oth sides of this last equation (ith respect to x $ and then solve the resultin# equation for y . hegeneral solution or 1quation ,.0 is

(here c is the

constant o


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E+a)ples


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E+ercise


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E+ercise


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E+ercise


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Bernoulli Equations

$ ernoulli dierential equation is an equation of the

form

(here n denotes a real number./hen n % , or n % 2$ a ernoulli equation reduces to a

linear equation. The su%stitution

transforms ,.3 into a linear dierential equation in theunkno(n function 4(x).


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E+a)ples


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0onlinear ODE

1-1st order ODE

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