Plan for today: Finish §1.4 1.Start §1.5 2.

Learning Goals: From §1.4, in addition to the ones from Lesson 6

Be able to solve problems involving population growth, radiocarbon dating and heating/cooling 1.From §1.5:

Be able to recognize a linear 1st order ODE when you see it, understand how it differs from types 1.of ODEs we have studied so far. Given a linear 1st order ODE, be able to compute an integrating factor for it 2.Be able to find the general solution of a linear 1st order ODE using an integrating factor 3.Be able to identify the largest interval on which an IVP for a linear 1st order ODE has a unique 4.solution (questions of this type appear often in the finals)

Reminders:

Read the textbook! 1.Fill out survey (link will be emailed shortly after class). 2.

Population Growth application of separable eaisSimplemodel Population where rate of births

is const in time rate of deathsis coast in time

PLA population at time t

f birthsperson in unit time

5 deathsperson in unit time

How does PCH evolve In time At

IP B P DE S P Dt

births deathsseparable

f 5 P DtAust fhIyoE PITH G 8 PCI 1 91

e

II y SIP ftp.fcp sldtlap p 51T t C

p ecek StW

pays poets at

Population at time t o

N B 8

fig PCHp f SO pas

Culture of bacteria at their numberincreased 3 fold in 10 hoursHow long did it take for ofbacteria to double assuming growthmodel above1 PCH Poe

KtKs P S

t in hoursP IO 3 Po increased 3 fold

pp ekw 3Po

k lo la 3 k butI 0

Looking for t Pct EP

Poletti 2P e left 2

thot luz t 4 lohours

Linear 1st Order ODE 1.51See defy fat

dff fangly separable

today dtd PITY 0no y

no y dependencedependence

Ex dt t x'y 3sinkdk

def 3costly x2squared

N tf 3xy2 cos Cx

sxy9119Egeylam

Note It 3xy both linearseparable

d cos Cx both linear

HowtosoLet pay

efPHdxObservation

yCx7 p'Hye teddy CH

P xJPdg efRx7dxy

so givenP Pcx7ytyJ

ddf tPCxy QCx7

hlqttiplythroughb.y pix e SPGDX By

pd Play p Qld

pay pGlQCxEa'u of type

p yad Jp Q dx t c

y fpcx1Qkldx t C

General Solis to F t Pay QQ

p ed

is called an integratingfactor

More generally an integrating factor isa fat you multiply both sides of ouredu by and it turns them into derivatives

Exits dff Y Lthe coefficientpix Qof y

Integrating factor p e JP deAdx

Exle'T eM e

Eff e 2xy ex x

fyfe y xe

e y face dx c

ex'y text c

y I Ce heneralsolin

Ex 2 Iff x'y cos

Integrating factory PLAY QCx

to bring into standard form

did yco

padESP dxPG e

3

Finishexa uple