Lecture H

Intro to 2nd order ODE's

Specialises:

2nd Order ODES.

i) dtmf,

= Flt ,dyq)

dhufi.FH.y.dz)let dy :

It

d£=FH

,

httstorder

Initial conditions : yltot Yo {

alto.

-

no{ jitosgi

⇒

usMostimportant example:

Than dy HIatNewton's

2MLaw9Motion: {

yttosyo t

mass x acceleration store ⇒

yltsyotfttslds2toMidas

Flt,x

, data) Ensample. Freefall withdt '

referee ainresistavd :

m

XmdIqs -

mg -8dgAt

Nyt : position andveloityat y 9- that :Farag

HI mfine -↳ uniquely determines AH.

lsmgsfgaog

(2) Autonomous 2nd order ODE's Example ( Harmonie Oscillator )- spiny spying

dfu.sftxihfu.IE#Agf?hw,

tomeeeen,

Phase plane air?µ⇒ .

Force = - km⇐ dIsva a

,i . I md7x At

Solution AAH'b

" AE- At I

dogs - kn,r a de rn{ vsHtt← ← ←

→ wife )→

,

> s

9ey idea : View w j Sh,

<

as a function of X r ← a

Conservation of EnergyIntegrate with respect to Ai

index s - boxIs fmvdrgxdxs - fbxdx

⇐ rmdrr

qts- tax ⇒ tgmv 's - lgkx 't E

.Yeon Stant of

Chain Rule : drfsdgq.gg,

⇒ Imu 't 'gkx2=E integration

So kinetic t potential = constant.

modify = - tax energy aagy

Check : dat ( Imr 't 'ska ) = mwdjtkxdxgt-f-kxywtfkndn.IO

Can Use conservation ofenergy

to Solve ODE.

tgmvtatkx 's E I constant ){

v .. ma

dxqst.VSIVE.ATSo '

gmfhfujttah.ME⇒ Gary⇒ I Hatt 's 2E#i yaeys.pt =±VImttC

Hat html 't - xD ⇒ sin " ( Hygeia)=±VkInttc

" xH=V2Esin(VImt -

p)A a

W 't 9- ft 's§ x # sV¥asNImt - pl sink

ask.tt

Linear ODES-

o C Homogeneous) L is an examplediff + pH2=1, # (nonhomogeneous)g alinear

d¥. tpkdyttqkty =/:*"ftnmwtheohgaoous

,operator:

Shorthand

notation

: Llc, y ,

# t fyakt]Last ftp.kijtgtay so

, Leg, HIT SKIED.

Exapte If Kyle y"t4y So yassin Htt 's a solutionThen Loni GTD of the ODE

= ( sink 44in httLaps get 42=0

= - 4 Sir htt t 4 sink -0

.

Theorem Seyne pH , GH,a

.net#.ae.mtnumson the

interval a stab.

If actual ,then the initial value

problemL[yJ=jIpHg 't

qttly-ttttyttd-goyttos.gs

Has a uniquesohtim, ya defined for all act ab .

Bad news : TheoremMain Example :

- The Cdamped ) harmonicdoes not give away to

find to

soliton!

oscillator.

8 be .

Good_news: If

padand

mulled ,

. md'xq= - kx -8dgglttsc( i. e constants) , so at

Lop = y"

t by 't Cg mdhfuettdxttkxsoThen there aye methods for

Solving the I. V. P

.

Plan for the next dweebs :

(2) Study the nonhomogeneousN Study the homogeneous case

case : Lay ] = fat

Upsay"

t by 't Cy = O in special cases :

.

Examples . Solve the IV. P.

LEDs y"t3g't2y=o glo ) -1 , y 'lok2

Solution :

Tert ]=ft3rt2)e*so⇐ r2t3rt2 so yw) = C. tf

⇐ I rt 2) Crt DsoI

gilds - g - as {⇒ rs -2 or rs - I

toyetis Get + seat⇒ 1cg's:3

is a solution for any 94£.

⇒ guts 4513 e-at