/6/008 Coious sysm: diffrial quaios Drmiisic modls drivaivs isad of (+)-( r( compar ( + ) R( + r ( (0) ( R ( 0 ) ( Dcid wha hav a ffc o h sysm Drmi whhr h paramrs ar posiiv or gaiv, i.. giv growh or rducio Drmi whhr h paramr is liar or oliar i rlaio o your iy (of populaio dsiy or amou of a compoud) Drmi h quaio bx birh rx growh Populaio x Populaio x i -dx immigraio dahs Wha ac o h sysm, iy: sig, cosa or liar/oliar rlaio: rx is posiiv, liar d r Wha ac o h sysm, iy: sig, cosa or liar/oliar rlaio: bx, posiiv, liar dx, gaiv, liar i, posiiv, cosa d ( b d) + i Drmi h quaio bx birh rx illväx Populaio x Populaio x i -dx immigraio dahs diffrial quaios, par Liar diffrial quaios Sparabl quaios Sysm of liar diffrial quaior chap.- Usig umrical mhods chap.4 Wha is diffrial quaios? Drivaivs i a quaio, i boh y ad y i h vry sam quaio Exampl: Boh cocraio ad chag of cocraio To solv his o how o mak som kid of igraio, ha is mak y o y. No: mos of h diffrial quaios is o possibl o solv aalyically, o is rfrrd o umrical soluios Sparabl quaios Spara variabls o rspciv righ ad lf hadsid RHS rsp LHS Thrafr igra ach sid: d d d
/6/008 Soluio, gral d d l l Soluio, gral ad paricular d Has h gral soluio: x ( If w kow ha, for xampl, 0)4 h w ca fid C i h paricular soluio 0) 4, 0) 4 0 Sparabl quaios, summarizig Spara variabls o RHS rsp LHS Igra ach sid r( ( r( ) ( K r ( ( K????????? ( + C' r r Irpr diffrial quaio How will h soluio, soluio fucio, look lik? Wh h drivaiv is gaiv h fucio dcras posiiv h fucio icras zro h fucio is cosa, a quilibrium sudy ( r( ) ( K Liar diffrial quaio: igraig facor Liar rgardig (, (rgardig h soluio fucio w ry o drmi) Ths liar quaios ca hc b wri as: ( Th Igraig facor is usd o solv hs kid of quaios. Igraig facor: µ( soluio by igraig facor ( igraig facor: µ( ( µ ( µ ( ( µ µ ( If µ( ) xiss (solvabl) you ca drmi a soluio for ( ( ( )
/6/008 Exampl: igraig facor ( igraig facor: µ( ( ( µ ( ) µ ( Exampl: + ( µ( soluio: ( µ ( ) ( ) C µ( Mapl: program ha fids aalyical soluios i(xp((/)*(^))*(*^),; Uss h sam hiqu as us, ha is a daabas. Bu is i h compur isad of i our hads. Exampl: igraig facor ( igraig facor: µ( ( ( µ ( ) µ ( Exampl: + 0.( 0. 0. µ( ) µ ( (0. 0.) 0. -0. soluio: ( (0. -0. ) 0. 0. Parial igraio: drivaig o rm, igraig h ohr Exampl: igraig facor + 0.( 0. 0. gral soluio: ( (0. -0. ) 0. 0. Exampl: (0)0. och Paricular soluio: Mapl: i(xp(*(0.*-0.),; (0) 0. 0 0. 0.-0.+C C0.4 ( 0. 0. + 0. 4 0 Exampl: Homogous liar diffrial quaio wih cosa cofficis Sam kid of problm as i liar rcursiv quaios, chapr. Fid h igvalus. Or rahr h roos of a polyom. d x d x dx a0 + a +... + a + 0 a x d x dx 0 Boh h firs ad scod drivaiv i h q. Eigvalus ad h characr of h soluio Soluio is achivd by assumig ha h soluios is: λ C λ Pu C i h quaio ad drmi λ d x d x dx a0 + a +... + a + 0, a x C λ λ λ λ a0 Cλ + acλ +... + a Cλ + ac 0 a0λ + aλ +... + a λ + a 0, iff C 0, λ 0 λ
/6/008 Eigvalus ad h characr of h soluio a0λ + aλ +... + a λ + a 0, om C 0, λ 0 This is a polyom, fid h roos! Exampl: d x dx 0 Iiial valus: 0), x () λ assum soluio x C λ λ 6 0, iff C 0, λ 0 x C λ, λ wih 0), x'() givs x Eigvalus ad h characr of h soluio d x dx 0 Iiial valus: 0), x () Paricular soluio: x For larg x soluio characr If Im(λ)0 R(λ)>0, xpoial growh R(λ)<0, xpoial dcras R(λ)0, cosa If Im(λ) 0 R(λ)>0, icrasig oscillaios R(λ)<0, dcrasig oscillaios R(λ)0, oscillaio Sysm of firs ordr homogous lijar diffrial quaios wih cosa cofficis Coupl a s of liar quaios, for xampl wo or mor populaios or agclasss or. Th populaios hav ffc o ach ohrs growh, a liar ffc. Almos lik agclasss bu h soluios is of λ isad of R dx ax + by dy cx + dy Eigvalus ad characr of h soluio x C v () y C v () a A c λ λ b d v () λ v () Drmi igvalus, λ ad λ as wll as igvcors, v ad v λ You mus hav a iiial valu (x,y) o calcula C och C Eigvalus ad h characr of h soluio Calcula igvalus, λ ad λ If h domiaig igvaluis posiiv boh x ad y icras rgardlss of iiial valu Th qilibrium, which is (0,0), is h a sourc If boh igvalus ar gaiv boh x ad y dcras rgardlss of iiial valu. Th qulibrium, (0,0), is h a sik S pag 60 for furhr dfiiio o mak from h igvalus (if complx c) Th characr of h soluio Grallly: for liar sysms all variabls ohr id cras or dcras xpoially xcp i fw xcpioally cass. Hc, a modl ha dals wih a sysm ha is robus ad sabl ough o b xprssd as a oliar sysm 4
/6/008 Mos quaios ca b approximad wih liar fucios ovr a shor irval quaios ha ar possibl o driva, ha is coious ovr a irval, ca b approximad wih a Taylor xpasio. Usig h drivaivs Th firs m of h Taylor xpasio is a liar rm ad hc i possibl o approxima Th irval is smallr h largr drivaiv umrical soluios Mos diffrial quaios is o possibl o solv aalyically. O hav o solv i by umrical mhods Eulr mhod is a way o mak a diff kv o a diffrc quaio wih appropria sp-lgh Wih mor rfid mhods, lik Rug Kua, uss a s of drivaivs a ach poi. I his way h dircio of h soluio is improvd for ach sp, Malab: umrical soluios [,y] od4( fucio',imirval,iiialvalus); Cra a m-fil rigid.m fucio dy rigid(,y) dy zros(,); % a colum vcor dy() y() * y(); dy() -y() * y(); dy() -0. * y() * y(); Solv h quaio o h irval 0- [,y] od4('rigid',[0 ],[0 ]) Malab: umrical soluios [,y] od4('rigid',[0 ],[0 ]) plo(,y(:,),'-',,y(:,),'-.',,y(:,),'.') Rmak rigid.m ad s diffr quaios Som kids of diff quaios, so calld siff quaios, should b solvd by ods( ) siff diff quaios