AP3114 Lecture WK10 Solving Differential Eqautions I

7/23/2019 AP3114 Lecture WK10 Solving Differential Eqautions I

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AP3114/Computational Methods forPhysicists and Materials Engineers

7. Solving diferentialequations I

Nov. 2, 2015


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Contents

Introduction to differential equations

Symolic solutions to ordinary differential equations

Applications of !"Es# analysis of dampedand undamped free oscillations


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Introduction to differential equationsDiferential equations are equations involving derivativeso a unction.

1. Ordinary diferential equation!en t!e de"endent varia#le is a unction o a single inde"endent varia#le,t!e diferential equation is said to #e an ordinary diferential equation $OD%&.

q ' ()*$d+d-&2. artial diferential equation

I t!e de"endent varia#le is a unction o /ore t!an one varia#le, adiferential equation involving derivatives o t!is de"endent varia#le is said to#e a "artial diferential equation $D%&.

2

-2

2

y2

-' 0.3. Order and degree o an diferential equation

+!e order o a diferential equation is t!e order o t!e !ig!est(orderderivative involved in t!eequation. +!us, t!e OD%

q ' ()*$d+d-&is a 4rst(order equation,


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+!e degreeo a diferential equation is t!e !ig!est "oer to !ic! t!e!ig!est(order derivativeis raised. +!ereore, t!e equation

$d3ydt3&2$d2yd-2&5(-y ' e-,is a t!ird order, second(degree OD%, !ile t!e equation

yt ' c*$y-&,is a 4rst(order, 4rst(degree D%.

6. inear and non(linear equations

8n equation in !ic! t!e de"endent varia#le and all its "ertinent derivativesare o t!e 4rstdegree is reerred to as a linear diferential equation. Ot!erise, t!e equationis said to #enon(linear. %-a/"les o linear diferential equations are9

d2-dt2 :*$d-dt& ;0*- ' 8sin$;t&,

and


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5. *$d2yd-2&2(5*y ' e-,

!ere all t!e coe=cients acco/"anying t!e de"endent varia#le and itsderivative are constant, ould #e classi4ed as a t!ird order, linear OD%it!

constant coe=cients. Instead, t!e equation2


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On t!e ot!er !and, i t!e rig!t(!and side o t!e equation, ater "lacing t!eter/s involving t!ede"endent varia#le and its derivatives on t!e let(!and side, is non(Bero, t!eequation is saidto #e non(!o/ogeneous. Non(!o/ogeneous versions o t!e last to

equations are9d2-dt2 :*$d-dt& ;o*- ' 8o*e

(tC,

and$-(1&*$dyd-& 2*-*y ' -2(2-.

7. Solutions

8 solution to a diferential equation is a unction o t!e inde"endentvaria#le$s& t!at, !enre"laced in t!e equation, "roduces an e-"ression t!at can #e reduced,t!roug! alge#raic/ani"ulation, to t!e or/ 0 ' 0. or e-a/"le, t!e unction

y ' sin -,


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Introduction to differential equationsE. Feneral and "articular solutions

8 general solution is one involving integration constants so t!at any c!oice ot!ose constants re"resents a solution to t!e diferential equation. ore-a/"le, t!e unction

- '


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G. Heriying solutions using S


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10. Initial conditions and #oundary conditions

+o deter/ine t!e s"eci4c value o t!e constant$s& o integration, e need to"rovide values o t!e solution, or o one or /ore o its derivatives, at s"eci4c"oints. +!ese values are reerred to as t!econditions o t!e solution. ore-a/"le, e could s"eciy t!at t!e solution to t!e equation.

d2

ydt2

y ' 0,/ust satisy t!e conditions

y$0& ' (5,and

dydt ' (1 at t ' 5.

Initial conditions are "rovided at a single value o t!e inde"endent varia#le sot!at aterevaluating t!ose conditions at t!at "oint all t!e integration constants areuniquely s"eci4ed.In general, 4rst order diferential equations include one integration constant,requiring only

one condition to #e evaluated to uniquely deter/ine t!e solution. +!us, t!ist e o e uations


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oundary conditions, on t!e ot!er !and, are "rovided at /ore t!an one valueo t!e inde"endent varia#le$s&. +!e ter/ J#oundary conditionsK is used#ecause t!e unction isevaluated at t!e J#oundariesK o t!e solution do/ain in order to s"eciy t!esolution.8n e-a/"le o initial conditions used in a solution ill #e to solve t!e

equationd2udt2 2*$dudt& ' 0,

givenu$0& ' 1, dudtLt'0' (1.

8n e-a/"le o #oundary conditions used in a solution ill #e to solve t!eequation

d2yd-2y ' 8sin-,using

y$0& ' 82, and y$1& ' (82.In general, t!e solution o a n(t! order OD% requires n conditions.


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y sy/#olic solutions e understand t!ose solutions t!at can #e e-"ressedas a closed(or/ unction o t!e inde"endent varia#le.

1. Solution tec!niques or 4rst(order, linear OD%s it! constant coe=cients

8 4rst order equation is an equation o t!e or/a*$dyd-&n #*y/' $-&,

!ere a, #, n and / are, in general, real nu/#ers. So/e s"eci4c tec!niquesor linearequations, i.e., !en n ' / ' 1, ollo. +!is catalog o solutions or linearOD%s is intended as

a revie o t!e tec!niques.

%quations o t!e or/9 dyd- ' $-&

8n equation o t!e or/ dyd- ' $-& can #e re(ritten asdy' $-&d-,

and a general solution ound #y direct integration,


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M dy 'M $-&d-,or

y ' M $-&d-


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er(de4ned unction intpoly9


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+!us, t!e general solution is 9

y$-& ' 2-0.5-20.75-6 <


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2. Solutions o !o/ogeneous linear equations o any order it! constantcoe=cients


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Su""ose t!at t!e c!aracteristic equation !as ninde"endent roots, t!en t!egeneral solution ot!e linear, constant(coe=cient, !o/ogeneous OD% o order ngiven earlier is

I out o t!e nroots t!ere is one t!at !as /ulti"licitym, t!en t!e mter/scorres"onding tot!is root in t!e solution, ill #e

%-a/"le 1? Deter/ine t!e general solution to t!e !o/ogeneous equation

In ter/s o t!e D o"erator, t!is OD% can #e ritten as


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+!e c!aracteristic equation corres"onding to t!is OD% is

+o o#tain solutions to t!is equation in S


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Applications of !"Es# analysis of damped andundamped free oscillations


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NetonQs second la, !en a""lied in t!e -(direction to t!e /ass / is rittenas9


!ic! results in t!e second(order, linear, ordinary diferential equation9

nda/"ed /otion

et us irst consider t!e case in !ic! t!e /otion is unda/"ed, i.e., : ' 0.+!e equation int!is case reduces to

corres"onding c!aracteristic equation is

it! solutions,


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result suggest a solution o t!e or/

ternatively, #y ta)ing

e solution can #e ritten as

!e quantity

is )non as t!e natural angular frequency o t!e !ar/onic /otion t!atresults !en no viscousda/"ing is "resent.


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a/"ed /otion


I da/"ing occurs $:R0&, t!e c!aracteristic equation #eco/es

!ose solutions are

!ere

+!e nature o t!e solution ill de"end on t!e relative siBe o t!e coeicients and ;o, as

ollos9

is real, and t!e solutions o t!e c!aracteristic equation are


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+!e solution to t!e OD%, t!ereore, is ritten as

!e "ara/eter

re"resents t!e da/"ed angular requency o t!e oscillation, and T1

re"resents t!e corres"onding angular "!ase. 80is t!e a/"litude o t!e

oscillation at t ' 0. I e de4ne a varia#le a/"litude,

t!en t!e solution to t!e OD%, also )non as t!e signal, can #e ritten as

lease notice t!at t!is solution is very si/ilar to t!e case o an unda/"edoscillation, e-ce"t or t!e act t!at in a da/"ed oscillation t!e a/"litudedecreases it! ti/e. +!e a/"litude decreases, or decays, it! ti/e #ecause

t!e "ara/eter ' :$2/&is "ositive. +!ereore, t!e unction e-"$(t& decreasesit! ti/e.


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I ' ;o, t!en t!e c!aracteristic equation "roduces t!e solution U' (, it!

/ulti"licity 2, in !ic! case t!e solution #eco/es

+!is solution re"resents a linear unction o t su#Vected to a decay actor,e-"$(t&.

I W ;o, t!en X$2

(;o2

& ' Y is real, and Y Z ,t!e solutions o t!ec!aracteristic equation #eco/e

#ot! negative. +!ereore, t!e resulting signal can #e ritten as9

Notice t!at t!e last to cases, na/ely, ' ;oand W ;o, "roduce signals

t!at decay it! ti/e. +!ese cases corres"ond to !ar/onic /otions t!at aresaid to #e over(da/"ed, i.e., t!e viscous da/"ing is large enoug! to quic)lyda/" out any oscillation ater t!e #ody o /ass / is released.


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+!e e-"ression or t!e "osition o a da/"ed oscillatory /otion is given #y

le t!e velocity, v$t& ' d-dt, is given #y

Fiven t!e initial conditions -$t0& ' -0and v$t0& ' v0, e can or/ a syste/ o

to non(linearequations in t!e un)nons 80 and T1, na/ely,

it! a""ro"riate values o t!e "ara/eters , ;I, t0, -0, and v0, e can use

S


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ample ? Da/"ed oscillatory /otion9

lot "osition, velocity, and acceleration corres"onding to t!e olloing"ara/eters9 / ' 1 )g, :' 0.1N*s/, ) ' 0.5 N/. +o deter/ine t!econstants 8oand T1, use initial conditions, -0' 1.5 /, and v0' (5.0 /s.

it! t!ese values,

Since, Z ;o, t!e resulting signal is t!at o a da/"ed oscillation it!

+o solve or t!e constants 8oand T1it! S


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Ne-t, e enter t!e )non values, and select a 4rst guess or t!e solution s09

+!e solution or s$1& ' 80 and s$2& ' T1 is o#tained #y using fsolve9


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+!us, 80' 7.12213@6and T1' 1.35G1GG7, and t!e "osition -$t& is given #y

%-"ressions or t!e "osition -$t&, velocity v$t&, and acceleration a$t& or t!is/otion can #e

entered into S


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+o "lot t!e signals -$t&, v$t&, and a$t& in t!e t(interval $0,30& use t!e olloingS


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ating "!ase "ortraits o oscillatory /otion

8 "!ase "ortrait or oscillatory, or any )ind o, /otion is a "lot involving t!ede"endent varia#le and one o its derivatives, or to derivatives o t!ede"endent varia#le. or e-a/"le, a "lot o velocity, v$t&, versus "osition, -$t&,

re"resents a "!ase "ortrait. Ot!er "!ase "ortraits ould #e a$t& vs. -$t&, anda$t& vs. v$t&.Example.

lot t!e ti/e(de"endent "lots and "!ase "ortraits or t!e signal o#tained inExamplein t!e "revious section.

+o "lot t!ese "!ase "ortraits e generate data on "osition, velocity, andacceleration as unction o ti/e t in t!e interval [0,G0\, as ollos9


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"!ase "ortraits are generated as ollos9


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$%&'

36

Due9 Ga/, Nov G, 2015ate "olicy9 10] of "er day

+o #e su#/itted online to


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E(ercise

1. Deter/ine t!e general solution to t!e olloing linear ordinary diferentialequations usingt!e corres"onding c!aracteristic equation9

Note9

_ecord your inter/ediate results using Jrint ScreenKl d # i ! ! ! l

2. lot t!e ti/e variation o "osition, velocity, and acceleration o a da/"ed/ec!anicaloscillator or t!e olloing "ara/eters9

lot v(vs(-, a(vs(-, and a(vs(v "!ase "ortraits o t!e /otions descri#ed a#ove"ro#le/.

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AP3114 Lecture WK10 Solving Differential Eqautions I