Rlidterceelpractocefoeeetious

⑨ we have to solve 2K - I - 2RF - O 2K ( t- 0.6f) so{-F to.arF=o

{ FC-ito.hr) -- o( t ) R - O and F-O ⇒ (0,0)

cu) R -o and - I to .9R=o ⇒ imposable

Ceu) to-GF- O and F-' O ⇒ lie possible

(w) 1-0-GF -- o and -Ito.9R=o ⇒ ( 40.9 i 40.6 )

Q@ Since fly) for yin Et , D is equal to zero only when

y -0 i f- Co >57=130 empties that fly) is positive ni co , D

and f- C - e. 5) = - 2<0 empties that fly) is negative in Eso)nfcy)

so fly, looks like #fYdyff

- fly)

fly) so in co ,D ⇒ thee solution yet, is increasing .

fcyldo ni El , o) ⇒ thee ooeeetvou y CEI is decreasing

^1. o source

✓

⑦ thee epueeebuiee ooeeetiou is you =D .

since -yet"

is

differentiable,the existence and ceeigueuess theorem heads

.

so So solution curves do not cross eade other.{

• At t - o we have gild > o - Yolo)↳ these 2 facts say that y ,# s Yott) =-D .

⑧ vectorfeddirecoioeefoeld

<- e- ← →→ →→

← ← ←→ → →-s a-← →

prob_eOn •

K th LK YS -3

/l l - I -2i.l://.se558T - i

=•

-3 - ⑥

proble€ needed = y t e't

- thee associated homogeneous equation is doted = y .

⇒ thee homogeneous solution - s ghltl - et

→ guess a particular ooeeetiou ypltl -- Cest

(cest )'s Celt t et 2ce2t=ce" test ⇒ ee s

a particular solution is ypctle ett

general solution yet) = Ket test

phobia adde = - Ey t et '-z)"

yen -0

We first weave thee y- linear term to the LHS

dyFtt Ey = ( t'-234

the integration factor is µcH=exp[f¥dt]= exp ( 2kt) = tMultiply thee equation by recti

t2 dyEt 2ty = tact

'-214

# ( tag) = tact's)4

Edge ft' Ct 3-274 at = J f- eitdeee f. fees to = ,

( this toA

a-t'-2 du -- Stoltwot t -

- o!

Eyal = if Cttzl to eeeotiel coudetiae is y so

I- o = ¥ ( i-235 t C C = I15

Ey = ⇐ZzP ⇒ yes =⇐⇒s15

I5t2probleee4@t6eepueCebueupoIeetsateye-n.o, I, ameforyyt-ysmymcreanugldecreosigcn.cl) t -

- t Tio://i.tt/FC-I,oj-t-#C-4-tI-t

=✓µ

yea sourceI

: ÷:poets-

y= source

7- "- 777

- y , co) =3 yzcl)=2 ygco) =)

^

F -

µ¥' Tt""F-Yz#-I

prob## let yet) be the amount of ueeuey attiene t

,.

and C thee rate of withdraws

dyIf

= O -05g - c

C') Coleen C = 1,000,

thee epceieeb were precut is at

0.05g - 1000=0 ⇒ y= 20, ooo

So if we want yet ) to never be zero,

we need

the initial value y co ) z 20, ooo

cu) ke general the epuieebuee point is y - 20CWe need to

,ooo 220C so that John can continuously

withdraw l the money . this means C E500

proto adde = y' t2y2tµy - feely)

the curve in the bifurcation diagram is the oei faeapeo

y ( y 't ay tr) - o ⇒ y=o and y'tag tee -- o

+ fry

ttx.tt#Dq,* *it

to determine thee arrows we look at a pants

• peso y - I feecy) = It2=320 arrows are pouting upward

= fee - 5 y - I feely)= 1+2-520 arrows are pouting downward

• pro y - - I feecy) =L so arrows are pouting upward

-

re - o y =-3 feecy) = -271-1820 arrows are pouting downward

to determine the bifurcation values we need to solve

fee Cy) - o¥zfuy↳ £+282 try - o{Sy-

tag tee -- O

2nd epuotiou see -3g'-ay . Plugging it into Hee est eg .

y'

try'- Sy'

-45=0 ⇒ y7y?o ⇒ yeo or ye - i

if y -o ⇒ µ=o if y .- -i ⇒ µ= - 3C

-D2 -4th - l

bifurcation values case µ- O and f - I

Additional

pr x2ad⇒ = y- xy yl-D= '

x' off = y Ci- x) yl dy =' doc

{ YI dy - f 'dx= J # -I da = -¥ - tubal t D

lulyl = - ¥ - label t D

(yl = e- I- he ''' tis

Igi , ed . e-I e- Gebel

y = te's e-¥ eke bei

- i

w-C

y Cx) -- Ce'¥ I I - ye-ne Ce

'

÷ = e- e ⇒ c-- E'

134

ycx) = e- t - YaLtxt

'

problee€Suede feet only depends out , the dopes ou eade vertical

eerie ate the same .

if we have one solution curve,then any horizontal

shift of this solution curve is still a solution curve

problee⇐ let yal be thee aueoaet of ueeuey cis bank at year t .

TIL = O-Olly t 1140 y Cole 400

probingCa) daff - sty - stet

'

integration factor µH-- exp ( f -atdt) = exp C - te)-

- Et'

e-Faded - stet} - St off ( e-"

g) = St

e-"

y -- Jstdt = It't c yak e" ( Et't e)

(b) dots = zytcoot .

honey. eg . offed = 2g ; a solution is ghetto e"

guess particular ooeeetoouy.pk )- A cost t Bseirt

- Aseiet + Boost = 2A cost + 2B suit t cost

eoeff Cost i B=2A ti B - D= 45

eoeff sent :I-A -2B

.

{ AI -IIB"

A= - as

Yp#=-€ cost t #sent

general ooeeetoou yes = kelt - ¥ cost t suit

(e) dyat

= ftet

t- hang. eeg adf.ee =y .

A solution is f- e

-

guess particular ooeeetoui gpcti - et et

(Cte ) 's Cte te cet + feet -¥tet e - I

⇒ ypk7= t et

general solution is yet)= feet ttet

Cd) dfg = sty't style III = y42Et3E)

gt dy = (2Et3t2)dt ftp.dy-J2ttstdt- YI = t't t

'to f- = -E- t' -c

yet) = _€EE

probleee diff = de'd -Ny -- faly )

to find bifurcation pts we need to solve

f-acyl - o Let- Ny ⇒ defeat{ Italy' - o tae- aso

Taeko -⇒ 2122g Icty)-o ⇒ 4--0 and yet{ des-azo2=0 ⇒ 2nd ep becomes 0=0

.✓

4--0

g.=L ⇒ old ep . beau de -22=0 L (e-a)to Ease

bifurcation pts ate 4=0 and die.