[PDF] MA 266, Spring 2021 Recap of §11-25, Recipe Book, and Problems

101373_6Review_1_solutions.pdf

MA26 6,Spring2021

Recapof§1.1-2.5,Recipe Book,andProblems

02/21/2021

Non-comprehensivelistoftopicscovered

Chapter1

•§1.1:Whatisadi !erentialequation?Whatisa solutionforadi!erentialequa- tion?Knowhowtoc heckwhether agivenfun ctionis asolutiontoadi ! erential equation.Orderofadi ! erentialequation,di ! erencebetweenor dinaryversus partialdi ! erentialequation.Whatisa nInitialValueProblemforafir storder di ! erentialequation? •§1.2:GeneralS olutions,ParticularS olutions.Beabletosolvedi!erentialequa- tionsofthe form dy dx =f(x).Beabl etocon vertbetw eenphy sicalunits. •§1.3:Slopefield s,solutionc urves.Beabletochec kwhethertheTheoremon existenceanduniqueness ofsolution sappliestoagivendi ! erentialequation (Theorem1in1.3). •§1.4:Whatisas eparabled i!erentialequation?Beable toidentifyonewhen youseeiti nNature("Na ture" meanse xams)andsolveit .Implicitsolutionsof di ! erentialequations,singu larsolutions.Populationgrowth ,Newton'sLawof

Cooling,radioactivedeca y.

•§1.5:Whatisal inear1sto rderOD E?B eabletoidentifyli near1stor derODEs inNatur eandsolvethemus inganint egratingfactor. Existencean duniqueness theoremfor1storde rlinearODEs .Knowthedi ! erenceintheassumption andtheconclusions bet weent histheoremandthetheoremonexistenceand uniquenessfromSection1.3.Mixture problems. •§1.6:Know theprocessofusing asubstitution tosolveadi!erentialequation. Beable toidentifyho mogen eousequations(inthesense dy dx =f(y/x))and Bernoulliequationsandsolveth emwiththeappropriatesu bstitutions. Beable toche ckwhetheradi ! erentialequationindi ! erentialformisexactands olveit. 1 !"#$%&"'()*+$,)-).!"#$%$/)0$11)2"34#")565$1521")74#4''40)89:;<= Beable tosolvereduc ible2n dorderdi!erentialequationswithth eappropriate substitutions.Beabletousenon-standardsubst itutio nsifneed ed,int hespirit ofExample6.

Chapter2

•§2.1:Beableto setupa di!erentialequationthatdesc ribespopulationgrowth. Beable tofindthepara meter sforthelo gisticequ ation,identifythecritical pointsandthecar ryingcapa city. •§2.2:Beable toiden tifyanautonomous equation.Beable todrawnandin- terpretaphasediagram.Stableand unstablec riticalpointsand equilibrium solutions.Logisticequationwi thharvesting,bifurcat iondiagrams. •§2.3:Beableto setupa ndsolveprob lemswhe rethemoti onofa bodyisa!ected byai rorfluidre sistan ce. •§2.4-2.5.BeabletoapplyEu ler'sm ethodandthei mprov edEuler'smethod toappr oximatesolutionsofagivenequation .Youshouldbeabletocarryout afewstepsofeachalgorithmbyhand.Beabletocomputetheerrorinthe approximationbetweenaknownexactsoluti onandtheEuler/Improved Eul er approximation. 2

Recipebook

Belowisasumm aryofthemost imp ortantentriesinourODEreci peb ookcovered inthes ections above. Pleasenote:theexampl esandnon-examples arenot necessarilymadeto beea sytosolve.

1.Antiderivatives:

dy dx =f(x)(noyontherigh thand side).(1)

Solvebyinte gratingbothsides withrespect tox.

•Examples: dy dx =cos(x), dx dt =t 2 . •Non-examples:y ! =yx,y ! =x+y.

2.Separable:

dy dx =f(x)g(y)

Solvebyse paratingvariables,i.e .writing

dy g(y) =f(x)dx,andintegratingboth sides. •Examples:y ! =yx, dx dt =tx+tsin(x) •Non-examples:y ! =x+y,y ! =xy+y 3

3.Linear:

dy dt +P(x)y=Q(x)

Solvebyfindingan integratingfac tor!(x)=e

!

P(x)dx

andreducingt o d dx (!(x)y(x))=!(x)Q(x), whichisoftheform (1). •Examples:y ! +xy=cos(x), dx dt =xsin(t)+ 1 t •Non-examples:y ! +xy 2 =cos(x),y ! +xy=cos(x)y 2 ,y ! +xy=cos(y).

4.Oftheform

dy dx =F(ax+by+c).

Usethes ubstituti onv=ax+by+ctoreduc eto

dv dx =a+bF(v),whic his separable.

Example:

dy dx = x+y+2 x+y"1 . 3

5.Homogeneous:

dy dx =F(y/x). Quickwayto identify:replacexby"xandyby"yinthee quationands eeif the"cancelout.

Solvebys ettingv=y/xandreducingto

x dv dx +v=F(v), whichisseparable. •Example:(y 2 +x 2 ) dy dx = x 3 +xy 2 x+y •Non-example:(y 2 +x 2 ) dy dx =x 3 +xy 2

6.Bernoulli:

dy dt +P(x)y=Q(x)y n .

Solvebyse ttingv=y

1"n andreducingto alinearequation. •Examples:y ! +x 2 y=cos(x)y 3 ,y ! =xy,y ! =xy+cos(x),y ! =sin(x)y 3 •Non-example:y ! +xy 2 =sin(x)y 3 Note:someoft heexamplesab oveare linearor separablesoitiseasierto treat themassuc h.

7.Exact:

M(x,y)dx+N(x,y)dy=0with

dM dy = dN dx onarectangle .

Solvebys etting#

x

F(x,y)=M(x,y),inte gratingtofind

F(x,y)=

!

M(x,y)dx+g(y),(2)

andsubstituting(6) into# y

F(x,y)=N(x,y)tofindg(y)andwithitF.

•Example:2xydx+x 2 dy=0 •Non-example:2xydx!x 2 dy=0

8.Reducible2ndOrder:Oneofthe following types:

4 (a)ymissing: d 2 y dx 2 =F " dy dx ,x #

Solvebyse ttingv=

dy dx .Thisreducesto dv dx =F(v,x).(3)

Solve(3),rec ove rv=

dy dx ,andsolveonemoredi ! erentialequationto recovery. •Example:y !! +2y ! =x •Non-example:y !! +2y ! +3xy=0 (b)xmissing: d 2 y dx 2 =F " dy dx ,y # (4) Solvebyregarding yastheindep endentv ariableandsettingv= dy dx ,which reduces(4)to v dv dy =F(v,y).(5)

Solve(5),rec ove rv=

dy dx ,andsolveonemoredi ! erentialequationto recovery. •Example:y !! +2y ! =9y •Non-example:y !! +2y+3xy=0 Note:Theresult ingequations(3),(5)mayormay notbeeasytosolve, butatle astthe yareof1storder. 5 y t2y gy Yy Bd uFy dx v I dy t w gy du Tey 9 2 homogeneous

Problems

Youwill probablyfinds omeoftheproblemsinthisl istchall enging.Doasm anyas youcana ndwewill discussth emincl ass.

1.ApopulationofPallascatsreliesonchanceencountersofmalesandfemales

forrep roductivepurposes.Thusthebirthrate$(birthspe runitoftimeper unitofp opulation)for thispopulationisan increasinglinearfunctionofthe populationitself.Thatis,$(P)=$ 0 +%P,where$ 0 , %arepositiv econstants. Moreover,thepopulationi srecoveringfr omadiseaseandtheirdeathrate& (deathsperunitoftimepe runitofp opulation)is decrea singwith time:itis of theform &(t)=& 0 +ˆ%e "t ,wheretismeas uredinmonthsand& 0 ,ˆ%arepositiv e constants. a.Findthedi!erentialequationthatdes cribesthepopulationgrow thofthe

Pallascats

b.whichofthefol lowing characte rizationsapplytoit? •Separable •Homogeneous •Bernoulli •Logistic •Linear •Autonomous 6 deafly outof timelcuuitofpop x x x AP Cp 8PDE T x births J x ly bee ofPY unituuitofport DP at f p logishcwodeeT.co tap fo de t P Bfo BIP po sode t P IP 8 80

Evariable

dd.PT BoBip SoP squared

2.Youare giventhef ollowingdi!erentialequations:

i. dy dt = " yii. dy dt +e t y=cos(t)iii. dy dt =e y iv. dy dt +e t y= 1 t+1 a.Forwhic hone(s)ofthemca nweguaranteethatt hereex istsaunique con- tinuoussolutiony(t)whichsatisfiesy(0)= 0andis defined forall tinsome smallinterv alaboutt=0? b.Forwhi chone(s)ofthemc anweguaranteethat theree xistsaunique con- tinuoussolutiony(t)whichsatisfiesy(0)=0andi sdefi ned forallt#R? 7 If etyt y I felt yl The two theorems pg GH

LGeneral

Ee

Unifueness

from 51.3
dy M M x flx.gl It e'y you yo thereisuniane f Dyf courtnear Xo Yo soda in somey interval containingxo 2Ex

Uniqueness

for linear eq's from 51.5
Ext Pang QCH Font onan interval whichcontains x then NPhas a unique soil inthe entiheinter So for a iiiii iv Inall these cases fHiy w fOyf con nearLordSoex uniquenessfollows fromfirst thin for bi ii only ii i'uarelinear Only ii has P Q cont in all of TR

3.Youare giventhef ollowingequatio nindi!erentialform:

(x 3 +yx "1 )dx+(y 2 +ln(x))dy=0.(6) a.Showthat(6)ise xact. b.Solve(6)withtheinitial conditiony(1)= 2. c.Supposethatyouwantt ouseanEuler 'smethodc alculatorsu chas https://www.emathhelp.net/calculators/differential-equations/ euler-method-calculator/ toapply Euler'smetho dwithstepsize0.1toapproximatey(2),whe rey(x) isthes olutionof (6)withtheinitial conditiony(1)=2.Wha tsh ouldyo u enterinthevari ousfield sinthelin ktoobtainthedesir edapproximatio n?

Whatisthe valueo ftheappro ximation?

d.Useyour answerfromp artb.andacomputera lgebrasystem tofindy(2) numerically.Compareyouranswerwitht heapproximationinpart c. 8 u n a Oyu IN GYM x t Q N xsoexact bLet f kest Qf x3tyx dyfy4lnx def Estyx fcx.gr yeux gig 4 2 dyf y thx lux egkyi y lux g Cgi 37C
So general solin is given by fix.y7 yeux 13 cco Want ye 7 2so 4 t 2hr1 t Ig c o c 3512
3 Sol xt E c For Euler write ddyy x'ty y lulx this willbe f Step size0.1 yl 17 2 Find yC2 0.573 d In plugin x2 use a CASto find y N

1,0329

SoEuler'smethod

isnot very accurate W step size h

0.01it

gives

1.0129

which iscloser

4.Acontainerwithcapacity100ltcontains60ltofasolution ofethyl alcoholwith

concentration50%andapopulationofbact eri a.Attim et=0asolutionof ethylalcoholwith concentration $ 80+
60
60+t
% %st artsflowingintotheco ntainer atarate of2lt/min.At thesam etimethew ellmixedsolu tionstartsflo wing outofthe tankata constantrate of1lt/min.Thebacteriainthetankwilldie oncetheconcen trationofthe solutioninthetankbecome s70%. a.Findthecon centratio nofethylalcoholinthetankaftertmin b.Dothebac teriadie beforethetankov erflow s? 9 too arin 2 n f routs1 N Xft iamount of ealcoholin etattimet dx It rinCin rout clout rind 2 too80 6607
I cHVoHH

6012TIt

Vol Ctl T T Foo 80
1,8 y initial rout60ft m fit

Ittsolro

foi.ee linear solve w integrating factor

Integratingfactor

p.ee ooItdt eluC6ottI Gott 460
tt xlt J 1.6 Gott 1 102dt
97.2
t t 0.8ft C SO f 97.2T
0.81 2 1 c Gott XCol 0.5

6030Lf

so 1800

Concattime

t Cf 97.2
t t 0.81 2 1800
Gott b

Bacteriadoe

when Ct 0.7 3 1800
t

97.2ft

0.81 0.7 f2 120
2 3600
0.1t2 t

13.2T720

0 tr41.5 min so they don't die sinceoverflowhappens atE40

5.Youare giventhef ollowingdi!erentialequation:

!x 2 sin(y) dy dx =xcos(y)+cos 2 (y)+x 2 (7) a.Useanap propria tesubstitutiontoreduce(7)toah omogeneousequation.

Hint:tryv=cos(y).

b.Bysolv ingthehomogeneousequ ationinpa rta)findanexplicitgeneral solutionfor(7). c.Findapar tic ularsolutionthatisdefinedinano penintervalcontainingx=1 (theope nintervalcanb eassmallasyouplease). 10 A vlxt cosly.CH du sinly Tx d

Substitute

x I x v tv21 2 dx bIsthis homogeneous replace V tv see if dcancel Ex Ex y xx f f f 2 122
2 1 homogeneous a I vUX II ut x ft thax X n't I f de Jd arctancu but'd 1C UH arctan E lulu 1C aretan WII lulxltc coslyxtan lukits y arecosx taulentxit c C y arccosxtau en mustbe inC l I tomake sense of arc cos taking c0would work sinceatx I we have xtanth 1 111
1 0

Reducible

u I y try x DX yIIEfEx t2v x linear p x et d e 2x et 2e 2 v e2 eve ev J xe dx e u fx dx x f dx x e C this gives u

Solvee

2 xe e 4 to find y