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18. 03 Cl ass 1, Feb 3, 2010

 I nt r oduct i on and f i r st met hods

[ 1] I nt r oducti on[ 2] What and Why[ 3] Separabl e equat i ons[ 4] Geomet r i c met hods

[ 1] Wel come t o 18. 03.

I hope you' ve pi cked up an i nf ormat i on sheet and syl l abus, and a pr obl em setwhen you came i n. Al l t hi s i s al so avai l abl e on t he web.

On the PSet you wi l l see r ef er ence to:

EP = Edwards and Penney, ( 6t h or 5th ed) . The very si mi l ar t o t he each ot herand t o t he 4t h. I ' l l t r y t o be sur e t o gi ve t he number i ng f r om bot h.

Notes = Not es and Exer ci ses. Avai l abl e f r om CopyTech i n the basement of Bui l di ng 11, t hough i t i s al so avai l abl e ( i n smal l pi eces) t hr ough t hecour se websi t e.

SN = Suppl ement ary Notes. Avai l abl e i n sect i ons t hr ough t he cour se websi t e.

 The f i r st PSet i s due Fr i day, Feb 12, at 12: 45, i n t he cubbi es at 2- 106,next t o t he UMO.

I al so hope you went t o r eci t at i on on Tuesday, wher e t hese yel l ow bookl et swer e handed out . More ar e avai l abl e her e. Usual l y I ask st udent s t o manuf act ur et he bookl et t hemsel ves, but t he UMO was ki nd enough t o do t hat f or youal r eady. Yea! We' l l use t hem f or a pr i mi t i ve but ef f ecti ve f or mof communi cat i on bet ween us. I t ' s pr i vat e; r eal l y I ' m t he onl y one who cansee the numbers you put up. Today, onl y one quest i on. More l ater .

I f you need t o change r eci t at i on, go t o the websi t e and f ol l ow t he l i nk t ot he "grade management syst em, " whi ch i s on St el l ar . The si zes are l i mi t ed.

 There ar e t wo l ect ure t i mes f or t hi s cour se, 1: 00 and 2: 00. You can at t endei t her l ect ur e, BUT you shoul d regi st er i n t he hour at whi ch you pl an t ot ake t he hour exams.

 The i nf or mat i on sheet and t he websi t e cont ai n l ot s of ot her i nf or mat i on.For exampl e, my OFFI CE HOURS ar e Wednesdays 3: 15 - 5: 15 : e. g. , t hi s af t ernoon.

Any quest i ons?

 You shoul d al so have pi cked up t he l i st i ng of t he 10 essent i al ski l l s atyour r eci t at i on. Teacher s of t hese cour ses know t he l i st of ski l l s. They expect you wi l l know how t o do t hese t hi ngs.

Her e' s a l i st of some of t he l ar ger cour ses l i st i ng 18. 03 as a pr e- r equi si t eor co- r equi s i t e.

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 2. 001 Mechani cs and Mater i al s I

2. 003 Dynami cs and Vi brat i ons

2. 005 Ther mal - Fl ui ds Engi neer i ng

2. 016 Hydrodynami cs

3. 23 El ect r i cal , Opt i cal , and Magnet i c Pr oper t i es of Mat er i al s

6. 002 Ci r cui t s and El ect r oni cs

6. 021 Quant i t at i ve Physi ol ogy: Cel l s and Ti ssues

8. 04 Quant um Physi cs I

10. 301 Fl ui d Mechani cs

12. 005 Appl i cat i ons of Cont i nuum Mechani cs t o Eart h, At mospher i c, and

Pl anet ary Sci ences

16. 01 Uni f i ed Engi neer i ng

18. 100 Anal ysi s I

18. 330 I nt r oduct i on t o Numer i cal Anal ysi s

20. 309 Bi ol ogi cal Engi neer i ng I I

22. 05 Neut r on Sci ence and React or Physi cs

Al t oget her 140 cour ses at MI T l i st 18. 03 as a pr er equi si t e or a co- r equi si t e.

[ 2] A DI FFERENTI AL EQUATI ON i s a r el at i on between a f unct i on and i t sder i vat i ves.

Di f f er ent i al equat i ons f orm t he l anguage i n whi ch t he basi c l aws of sci ence ar e expr essed. The sci ence tel l s us how t he syst em at handchanges "f r omone i nst ant t o t he next . " The chal l enge addr essed by t het heor y of di f f er ent i al equat i ons i s to t ake t hi s shor t - t er mi nf or mat i on and obt ai n i nf or mati on about l ong- t er m over al l behavi or .So t he ar t and pr act i ce of di f f er ent i al equat i ons i nvol ves t he f ol l owi ngsequence of st eps: one "model s" a syst em ( physi cal , chemi cal , bi ol ogi cal ,economi c, or even mat hemat i cal ) by means of a di f f erent i al equat i on;one t hen at t empt s t o gai n i nf ormat i on about sol ut i ons of t hi s equat i on;

and one t hen t r ansl ates t hi s mathemat i cal i nf ormat i on back i nt o thesci ent i f i c cont ext .

Sol veDi f f er ent i al Equat i on: _____ _____ _____ \ Behavi or over t i meShor t t er m i nf or mat i on /

/ \\ /\ /

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Model \ / I nt er pr et\ /\ /

\ /

Physi cal Worl d

A basi c exampl e i s gi ven by Newt on' s l aw, F = ma. a = accel erat i on,t he second der i vat i ve of x = posi t i on. For ces don' t ef f ect x di r ect l y,but onl y t hr ough i t s der i vat i ves. Thi s i s a second order ODE, and wewi l l st udy second order ODEs extensi vel y l ater i n t he cour se.

[ 3] I n t hi s f i r st Uni t we wi l l st udy ODEs i nvol vi ng onl y t he f i r st der i vat i ve:f i rs t order: y' = F( x, y) .

Exampl e 1: y' = 2x Sol ut i on by i ntegr at i ng: y = x 2 + c.

Not i ce t hat t here ar e many sol ut i ons.An expr essi on l i ke t hi s, i nvol vi ng a const ant ( c her e) i s cal l ed t heGENERAL SOLUTI ON. The const ant i s a "const ant of i nt egr at i on. "

Exampl e 2: y' = ky. General Sol ut i on: y = Ce {kx} . MEMORI ZE THI S

- i t ' s t he cent r al exampl e i n t hi s cour se.

( I n f act, a good def i ni t i on of t he exponent i al f uncti on e x i s t hat i t i st he sol ut i on of t he di f f er ent i al equat i on y' = y such t hat y(0) = 1 . )

Q1: What i s t he gener al sol ut i on t o t he ODE dy/ dx = 2y+1 ?

1. y = Ce {2x} - 1

2. y = Ce {x/ 2} - 2

3. x = y 2 + y + c

4. y = e {x/ 2} + C

5. y = Ce {2x} - 1

6. y = Ce {2x} - 1/ 2

7. y = e {2x} + c

8. None of t he above

Bl ank: Don' t know

 The met hod: "separ at i on of var i abl es. " You st udi ed t hi s i n r eci t at i onyest er day. Recal l t he met hod: Put al l t he x ' s on one si de and y ' son t he ot her ( i f possi bl e) :

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 dy/ ( 2y+1) = dx

I nt egr at e bot h si des:

( 1/ 2) l n| 2y+1| + c1 = x + c2

Amal gamate t he const ant s and ( i f possi bl e) sol ve f or y i n t er ms of x :

l n| 2y+1| = 2x + c , | 2y+1| = e c e {2x} , 2y+1 = C e {2x}

y = C e {2x} - 1/ 2

So the answer i s: 6

We can check t hi s! : y' = 2 C e {2x} = 2( y + 1/ 2) = 2y+1

I t ' s a ni ce f eat ur e of di f f er ent i al equat i ons i n gener al : i t ' s easy t ocheck your answer !

Q1. 2. I s y' + xy = x separabl e?

1. Yes

2. No

Bl ank: don' t know.

Wel l , y' = x - xy = x(1- y) so dy/ ( 1- y) = x dx : YES.

We coul d go on to sol ve t hi s, but you can do t hat on your own.

We wi l l see many ot her methods of sol vi ng var i ous t ypes of equat i ons.Unf or t unat el y, most r eal l i f e equat i ons ar en' t expl i ci t l y sol vabl e,and of t en you don' t act ual l y car e as much about t he expl i ci t sol ut i onas about t he gener al pr oper t i es. That ' s one pl ace wher e t oday' s t opi ci s hel pf ul .

[ 4] Gr aphi cal appr oach

 The ODE y' = F( x, y) speci f i es a der i vat i ve - t hat i s, a sl ope - at everypoi nt i n t he pl ane. Thi s i s a DI RECTI ON FI ELD or SLOPE FI ELD.

Eg y' = 2x : I dr ew some of t he di r ect i on f i el d. Not i ce t hat t he sl opeF( x, y) does not depend on y her e: I t i s i nvar i ant under ver t i calt r ansl at i on.

A SOLUTI ON of t he di f f er ent i al equat i on i s a f unct i on whose gr aph has t hegi ven sl ope at every poi nt i t goes t hr ough. I dr ew some. The gr aphs of sol ut i ons are I NTEGRAL CURVES. They are ver t i cal l y nest ed parabol as. The t r ansl at i on i nvar i ance of t he di r ect i on f i el d i s r ef l ect ed i n t hef act t hat a ver t i cal t r ansl at e of a sol ut i on i s anot her sol ut i on.

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 To speci f y a par t i cul ar sol ut i on of an ODE you have t o gi ve an I NI TI ALCONDI TI ON: when x t akes on a cert ai n val ue, y t akes on a speci f i edval ue.

Eg y' = y : I dr ew some of t he di r ect i on f i el d. Not i ce t hat t he sl opeF( x, y) does not depend on x her e: I t i s i nvar i ant under hor i zont alt r ansl at i on.

Gr aphs of sol ut i ons are now hori zont al t r ansl at es of each ot her .

Exampl e 3: y' = y 2 - x.

 Thi s equat i on does not admi t sol ut i ons i n el ementar y f unct i ons.Never t hel ess we can say i nt er est i ng t hi ngs about i t s sol ut i ons.

 To draw t he di r ect i on f i el d, f i nd where F( x, y) i s const ant , say m . Thi s i s an I SOCLI NE. Eg

m = 0 : x = y 2. I dr ew i n t he di r ect i on f i el d.

m = 1 : x = y 2 - 1

m = - 1 : x = y 2 + 1 .

I i nvoked the Mathl et I socl i nes and showed t he exampl e.

I dr ew some sol ut i on cur ves. We have seen i n act i on t he

EXI STENCE AND UNI QUENESS THEOREM FOR ODEs:y' = F( x, y) has exact l y one sol ut i on such t hat y(a) = b ,f or any ( a, b) i n t he r egi on wher e F i s def i ned.( The sol ut i on y(x) may onl y exi st f or x near t o a . )

( You actual l y have t o put some t echni cal condi t i ons on F - - see EP. )

 The E and U t heorem says t hat t here i s j ust one i nt egral cur ve t hrougheach poi nt :

EVERY POI NT LI ES ON J UST ONE I NTEGRAL CURVE: NO CROSSI NG ALLOWED.

 The appl et makes i t l ook l i ke many sol ut i ons coal esce, but t hi s i s j usta pi xel pr obl em. I n r eal i t y they ar e separ at e, but ver y cl ose t o each ot her .

Many seem t o bunch up al ong t he bot t om

br anch of t he parabol a. Can we expl ai n thi s?

I cl eared t he sol ut i ons and dr ew i n j ust t he i socl i nes m = - 1 and m = 0.Once a sol ut i on gets bet ween these t wo parabol as, i t can never escape. The poor t hi ng can' t cr oss t he m = - 1 parabol a because i t woul d havet o have sl ope gr eater t han - 1 when i t does; and i t can' t cr oss t he m = 0par abol a because i t woul d have t o have sl ope l ess t han zero when i t does. Thi s i s a FUNNEL.

So sol ut i ons i n t hat r egi on st ay i n t hat r egi on: t hey ar e tr apped

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bet ween t hose t wo parabol as, whi ch are asympt ot i c as x - - - > i nf i ni t y.Al l t hese sol ut i ons become ver y cl ose t o t he f unct i on - sqr t ( x)f or l ar ge x . Thi s i s an i deal si t uat i on! - we know, appr oxi mat el y,but wi t h i ncreasi ng accur acy, about t he l ong t er m behavi or of t hesesol ut i ons, and t he answer doesn' t depend on i ni t i al condi t i ons ( as l ongas you ar e i n t hi s range) . Thi s i s "stabi l i t y. " ( Of cour se i f t he sol ut i ondoesn' t get t r apped, i t ' s a di f f er ent story. )

Di r ecti on f i el ds l et you vi sual i ze t he qual i t at i ve behavi or of sol ut i onst o di f f er ent i al equati ons, and t hi s i s of t en what you want t o know.But we al so want t o be abl e t o sol ve ODEs "anal yt i cal l y, " t hat i s,usi ng f ormul as.

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18.03 Differential Equations �Spring 2010

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