Studi qualitativi per problemi di Cauchy

(1)

Studi qualitativi per problemi di Cauchy

Riccarda Rossi

Universit` a di Brescia

Analisi II

Riccarda Rossi (Universit`a di Brescia) Studi qualitativi Analisi II 1 / 210

(2)

Es. 1.

Dimostrare che la soluz.

( y ⁰ (t) = sin(t + y ² ) y (1) = 0

esiste ed `e unica, e che il suo dominio massimale di definizione `e R.

(3)

(4)

(5)

Es. 2.

Studiare al variare di ↵ 2 R esist. locale per ( y ⁰ = p

³

y t

y (1) = ↵

Gw fctiytfft

f ^e ^C°CD2)

^-

^Audio ^be

sudden

abilities vsfeeto

^a

^y

^.

(6)

¥yAyt{ ^FI

's sety

to set

.-y

fimfakr

^: ^eunsdeo

Ho

_,

_yo )

^con

Yo=to

^.

2fy ( ^to ^.to ) Ikbdmltfttotothtfeto ^.to ⁾

h→o

^- ⁼

he

(7)

±

Gmo moth ^-3¥

=

layoff

⁼ ⁺ ^a

_jj

wndi Fy ÷ ^esrste =LTy±Y}

solo

su

1122

-

D

,

con

D= { kyKR4Y=t }

(8)

Jnoltu

it D

otceooeis

fmnmwmmt ^> _t ng

Qwndi

:

Peter

^.

F ! locale ^si applied

chicanery

^con

^leg ^condition

Ylsqtydy

.

⇐D Hoieott

^,

²⁾ END

A a * ^s

(9)

Es. 3.

Determinare l’intervallo massimale di definizione per le soluz. di ( y ⁰ (t) = t + p

y ² + 2 y (0) = 1

( y ⁰ (t) = 1 t 1 +

p y ² + 2 y (0) = 1

b.)

Tp

^.

^del ^ten

^.

^di ^Z ^! ^locale _fee (a)

far yttt ^a ^E ^Cock ⁾

e acrid

¥y ^e- ^been ^defe ^ant .sn#perke/y2t2 ²² !

I _argon

^.

M

^non ^st ^ANNULLA

_! _!

(10)

ok ten

.

H

.

locale

Tpot

^.

^de ^tea

^.

^It

^.

^globale

-

:

domlft 1122=112×121 ^stnrsua _vertyole

oresuta sublime

Zk

_,

.kz ^to

^:

tct

,

y )

^c-

RZ

Ifk _y ⁾ ^I ^E ^kr tkzcyl

(11)

fctiyk ^tt ^# luecrescitasnbei

^.

imy

. non

sntult R2

,

me su

EMM ]xR tm _>

_o ,

%faHt

,

tft

,

g) ^EEMM ^] ^xD she

' fan

'I¥¥iFr±F¥I¥ ^:O

E Mtlyl

⁺

^✓2

^-

(12)

Allow ^R ^ten

^.

^di Z _! globalesi

applied ^a f- ^wsiuua

^a

EM ,m]xR

,

ttm >

o

.

→ Llintewallo

^mass

^male ^di F ^e-

[

Twin

_,

^Tmax _]= ^EMM ^] .t±

/

^I

M ^M

(13)

Per _l

^'

arbitrate

^'

oh

^'

M

_,

^R

domino

^mass

^male ^di ^F ^e- ^R

pb

^.

^di ^Cauchy ÷

µ

¥4212 y

=

^lot -÷

⁼

^fetiy ⁾

dom I f)

⁼

( ^R

^'

^hs _{} )} ^xD

^-

^In ^ualta

,

(14)

negli ^enunciate

^.

^dei ^teoz

^.

^srconsrdoa domlft ^I

^×

Y

_, ^on

I imtewallo I 7 to ( ^drove ^e- date

= )

Condit

.

du

hendry ⁾

^:

Qw

^:

I z ^O

,

I intervallic R 't }

→ 1- _to

_,

1)

^-

^Quinon

^,

^di ^falto

consider _dorm ₍ ft ^to

^,

1) ^xD

(15)

.

domlflisitnsua ^whale ⁽ Y=R ⁾

.

per apperaere ten

.

Z globed

_,

^demo

controller

so

f ^he ^cresuta ^subeinaere

in

Y

^.

Qw

:

If 1<-1 g) # #

⁺

^ME

teta # ¹

# ⁺ lyl

⁺

of

(16)

Per

_avere

create subeineore eimy

_,

den ^Shimane ^1-

^con

^wsiante

^.

the

Qnimdi

non ^mi

possoaauvicimorenaz

^.

aaa consider f mate a

too

,

b]x ^R

,

con bc ₁

,

white

No .

tltiylcfa

,↳]xR

^, ^.

Ifk _g) ^Is

# ⁺ ^lyle ^R ^°kdfmIFme

(17)

Meteor

^.

^di ^{F !} global ^appbcato ^a

£1 _to

,

b) ^xp imblrce ^eke ( Twin

,

Tmax ]= ^to

^,

^b ] tbcz

= b ¹

Seeome be Arrsiirapeo

,

re dominic

massmale e- Cis

_,

1)

(18)

(19)

Es. 4.

Verif. che la sol. di (

y ⁰ = (y + 1) ³ sin(y ) y (0) = ¹ ₂

`e limitata e strettamente crescente.

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

Es. 5.

Studiare, al variare di ↵ 2 R, la monotonia di sol. di ( y ⁰ = arctan(t)e ^y (y 1)(y 2)

y (0) = ↵

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

Es. 6.

Dimostrare che la sol. di

( y ⁰ = (y 3) arctan(log(y ² + 1)) y (0) = 1

`e definita su tutto R e calcolare

t! 1 lim y (t) e lim

t!+1 y (t).

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

Es. 7.

( y ⁰ = (y 2) arctan ² (y ) y (0) = 1

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

Es. 8.

Dato (

y ⁰ = e ^y

²

+ t ⁴ y (0) = 0

(1) Dimostrare che esiste un’unica soluzione definita su tutto R (2) la soluzione y (t) `e dispari?

(3) Calcolare lim

t!+1 y (t).

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

Es. 9.

Dato (

y ⁰ (t) = arctan(t) (y (t) arctan(y (t))) y (0) = y 0

studiarne la soluzione al variare del dato y 0 2 R.

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

Es. 10.

Dato il problema di Cauchy

( y ⁰ = arctan(y ) y (0) = ↵ dimostrare che la soluzione

1

`e definita su tutto R

2

`e strettamente monotona per ↵ 6= 0

3

`e convessa se ↵ > 0, concava se ↵ < 0

4

calcolare

t ! 1 lim y (t), lim

t !+1 y (t)

(92)

D

^•

fctiyk ^octane g) ^c- ⁽

^•

^C ^TE )

.

Ok F ! locale

.

dom

¹

f ^1=112×112

sTwseiavdhiele.cresutaosublrmeauimy.7@kTlfCtry11-laruaney1su7zVLtiy1cyR2-sFl.globaleok.CTmrm.T

maxttk

(93)

2) ^Soluz

^.

^STAZ

^:

^YIHEC

_,

^done ^CAR

e- tale actom ( CKO

ct

ho ! ^sdnz

^.

^sinz

^:

YHIEO

^.

Tlishnguo ^I ^aesi

/&=y(

^o

⁾ ^> ^o

\ Leo

(94)

Case HI ycd ^> ⁰

^.

•

per R ten

^.

di confront Are

²

^olnz

^.

boeali

,

to YH ^> ^o ^the R

→ y ^' ^At

⁼

aretomey ^LH ⁾ ^> ⁰

→ y ^e- ^shut

^.

^crese

^. ^su

^R

^.

(95)

link ⁾

Per mouotowa

,

F _L

- =

trna

_.

^Yen

^,

^L

^.

^E @ d) F ^Lt

⁼

beamy ^at ^, ^Lt ^EH

^,

^to ]

Per _calendar _L

.

,

Lt ^user ^tar

^.

dell

^'

_asrmtoto

_.

(96)

Appbu ^ten

^.

^esntolv ^for ^t→

^- ^a ^:

L

-

ER

,

Z lfjm

.

y

^'

^' ^H= fd _;

_→

onetomcyity

iaraom ( L

-

)

aora

,

arctan ( L

-

)

⁼

^O

→ L

^-

to

(97)

Per ^t

^→ ^to

, dimostw In L+

⁼

^to

( _Lt ^E ( ^a

,

to ] )

^-

-

a assurolo

_,

sra L ₊ ^E R

.

Src _come

bm

^to

÷ ^Y

^'

^'H= ^aruom ^C ⁴⁾

^,

ten ^M ^ten

^.

^asmhoho _,

Arian ( _↳ ) ^dourebbe ^esse ^re _=D

→ 4=0

^.

Assmdo

,

isto eke

(98)

Gnuessnte ^/ ^Concanbi

- ( ² ^>

^o

)

y ^' 'tH=§+

¹

^y ^'H ⁾

=§± ⁽ ^arciomiyla ⁾

→

=

Yttl

.µ@, ^> ^Ott

8rem

⁽

_y "l= srgmcyi ⁾ ye

^'

^convene

→ y

^'

^'tH ^> ⁰ ttepss

^-

(99)

I

•

per ^L

^< ⁰ ^:

y ^decree ^ante _,

L

^- ⁼

⁰

,

L

₊ = ^- a

y ^coneava

^sn

^RI

(100)

(101)

(102)

(103)

(104)

(105)

(106)

(107)

Es. 11.

Al variare di 0 < ↵ < 2⇡ studiare la soluzione di ( y ⁰ = 2 sin(y )

y (0) = ↵

•

fltiyt ² ^srniy ⁾ ^E ^C

^•

⁽ ^R ⁾

^-

-

0kt ! locale

.

-

domlft ^112×112

,

lfltiy ⁾ ^1<-2

Ok Z ! globale

^.

→ ttxer

,

Ctnwm

_,

Tmaxt ^TR

(108)

.

Sol

^.

^staznnorie

^:

ylH= ^kit , ttktk in parttcoleerei

-

Lme ⁰

^e

^2T

ho ³ soluz

^.

STAZ

:

YLHEO

^←

²⁼⁰

YLHEI _← _L=E

YHIEZI

_←

_L=2I

(109)

IIs

^'

^0<2 ^at ⁽ ^A ^Casa ⁾

IT

<

&

<

ZI

.

I

<

yc ⁰¹⁼²

^a

^2T

*

a <

YIHCZI ^HER 44+1=2 _srmiyltl I ₎

^- ^O

D Y ^I ^shut

^.

^deeresante

^su

R

-

(110)

2. ^3ft

^- ^q^i.

t* I

^- ^a

(111)

E A

^,

^2T ]

Per mouotowa

,

F L

-

=

ftp.YH

-

LEEK 'd z ^t

^→ ^to

^yih

^.

Per ten

^.

osntoto

, ( es

^:

dettagli ^! )

L

-

=

Zit

,

L+=I

(112)

yhltk ^d- ^/ YIHH

⁼

da ⁽ : ² ^sincylti ⁾ )

⁼

=

y

^'

^IH ^I cvslyltl ⁾

Se _de (a) ^at ) _, _y

^'

¹⁺¹

^<

^otter

→ srgm ⁽ y ^" CHI

=

-

Srgm ⁽ _eoslyct ⁾ )

(113)

Ricard

as ,z , { { ⁷⁰ 0 F ^Eta ⇐ E3zI ^#

Qwmoli

¥ { ⁷⁰ ^£0 ^se ^se ^Is ^zKyH< _yH<3zI ^at

(114)

Y ^shut

^.

^decresaute

_,

pwmdi

Z _! t*eR ^taleehe

ylt*l=3zE

e

per t<t*

,

YCH ^> § ^'T

ten ^t ^> ^to

,

YH ² 3zI

(115)

⇒ y ^I ^WNUNA

^sn

^to

^,

^t* )

Y

^e^'

^WNVESSA ^su ⁽ ^to ^.to ⁾

{ _t.ae ^ou _fees

^.

_pay

^'

(116)

(117)

(118)

(119)

(120)

(121)

(122)

(123)

(124)

(125)

Es. 12.

Dimostrare che le soluzioni di

( y ⁰ = e ^y sin(y ) y (0) = ↵ sono definite su R per ogni ↵ 2 R.

f ⁽ _try ^l= eudsrmey ) ^{E C}

^•

CRY

⇒ ok Z ! ^locale

^.

dm ( ft ^Rx ^ID

Mafctiyl

^e

_Ysimiy ) NI ^he ^creseibe

subhneare

ⁱⁿ

g

^.

(126)

Sospefdi ^he _question

^an

( Tomin

,

Tmaxl

.

GWZ

^.

_Statwnsree _- fctryteysrm ⁽ _y ₎

⇒

^A

soluz

^.

_state

,

yltt ^kit _, HKEZ

(127)

3 ^I

÷

T

:

T

- ZE

(128)

.

Se _L= kit

_,

per ^KEZ

,

→ ylttke

,

tt ER

.

Altnmeuti

- ,

d

^'

_sicuro

Z ! ^tee ² ^tale ^eke

[k⇒I< ² ^< EI → CEDEYIH ^LEE

(

^ooe ^'

_decompress

fne

2 multiple ^iuteii ^die )

(129)

CEDE ^< YH ^< ^Ee

y

tttc ⁽ Tmim ,Tma×)

L ^' ipoltsr ^di ^cresuta sublime _are eeueufruate evngo ^be ^sduz

lfkiyttyl

⁼

^is ^# lsrmiy AD Is @

(130)

→ ( Town

,

Imax )=R LIVER

Quedoueyronam fnonme ^tutte

6 ^volte ^eke ^wesw

^a

pwvare

one zm ^> o

^:

the ( ^Tmim

_,

Tommy

lyttle ^M

(131)

(132)

Un risultato di prolungamento

Nelle ipotesi del teorema di esistenza e unicit`a locale, se u `e una soluzione del problema di Cauchy e K ⇢ A un compatto tale che

graf(u) ⇢ K ,

allora il dominio di definizione di u non `e massimale.

M

^- ^→

^quest

^fe

^e-

^veo

•

domiuttaib _]

.

F M >

o ^:

tttetab _]

,

luutl

^±

M

(133)

Es. 13. (Tema d’esame del 14/01/2013)

Dato il problema di Cauchy

( y ⁰ = t ^{4 y} _y

²

y (0) = y 0

determinare al variare di ↵ 6= 0 se il problema ammette esistenza e unicit`a locale e globale. Studiare monotonia, eventuali simmetrie della soluzione, e i limiti agli estremi del dominio di definizione.

•

%

fetiyht

^.

⁴⁼⁴²

domlfk ¹¹²

^x

y ⁽ ^Ryo ^} ⁾

N0NEsRcSUAvERtcA=

(134)

fe ^est ^Rx ⁽ ^R ^'bo} ⁾ ⁾

^-

^OKZ ^!

.

locale

.

° non

posso tyoto

apphhereteor

^.

^It

.

globale

^.

homos ^se ^C ^Town

,

Tmax 7=121

.

(135)

.

flhZ

^.

STAZNNARLE

- ^:

fctiy ⁾

⁼

^t

^'

kg2=o ^a ^y=±2

Ctt )

I _soluz

.

STAZ

^.

yite ² ( yo=2 )

yttk ^-2 ⁽ _yo=

^-

2)

(136)

hats ^Y NI ONII

flat

.

STAZION

;

-

Yo eivalouproibito

2

per ya

.:

yo : I

yo

(137)

D. _go

^c-

²

2)

^.

Yo ^€f2

^, ^o

) 3)

^.

_Yo ^E ⁽ ^0,2 ⁾

4)

^.

yo ^> ^z } ^esenizio

(138)

D _40€ _YHE ^-2 ^e- ^al

^.

^FAZ

^.

a the ^ttectmim ^,Tma× ⁾

Y 'Htt

^.

hii

yttt Us °

> o

→ sign ⁽ y 'tHl= srgnlt ⁾

^.

(139)

⇒

see ( ⁰

^,

^Tmax )

, y 'H>o⇒

ye

^'

^Crescent

Su ( ^Tmim ^,o ⁾

_,

ytttcood

y decreseeute

2 *

(140)

Animals

^'

^ttte ⁽ ^Tomin

^,

^Tmax )

,

% .< YH ^a

⇒ ( ^Tomin

_,

Tma .×)=R ^I

venifee ^che ÷ _YH _EPARL

,

(141)

Got

,

into deeamdo

2- itty ^tt ⁾ _,

srvewfie ^die ^Z ^resolve

•

Zl

₌

t

^.

4-2-2

{

_.

_zcot _yo ^F

→ zctt YH !

⇒ yttkyttt ^Voter

(142)

.

Srecome y ^I ^PARI

^,

^be ^studio ^solo

he [ ^ates )

.

Rei monotone

,

Z him yH

^-

⁴

t

→

to

y crescssyodt ^E

^-

²

Con dl teen

^.

Asimoto

^:

L

₊ = ^-

2 ( ^per ^PARUTA

^' _,

L

^- ⁼ ^-

2)

(143)

-

z< yo

:

-2 < ytH< ⁰

I ^< ^o

Y

' 'H= t

^.

G=€ _yen ^> ^°

< o

signly ^'t

^-

_srgnlt )

^.

(144)

Y ^deerescesu ⁽ ⁰

^,

^Tmax ⁾

y ^cresu ^su ⁽ ^Tmrm

^, ^o

)

*

⇒ -z<ym< yo ttcctmim _,Tma× )

⇒ Tmm=→ , Tmax=tes

(145)

• So grot ^che _y ^I ^PARI

.

per ^r ^ten

^.

asntoto

,

uedw

^'

cke L ₊

=

-2

(146)

(147)

(148)

(149)

(150)

(151)

(152)

Es. 14. (Tema d’esame del 10/04/2011)

Studiare il problema di Cauchy

( y ⁰ = log log ² (y ) + ¹ ₂ y (0) = y 0

con y 0 > e ^1/ ^p ² .

efttiyhlogclogicyltied

^.

domcfk ^Rx ⁽

⁰^, ^to

⁾

↳

^none ^' ^una

sina.yanthf.d.ae ^Vertrudel

^.

^Z ^! ^Grosse

feet ^CRX ^catch ⁾

^⇒

^ok ^ten

^.

^F ^! ^locale

(153)

Qwmd

^.

Kyo ^> ^e- HE _he _sduz

. e-

def

^su

Amin

_,

Tmax )

,

per ^old ^now

speufreto.SoluZ.stAhoNARrElogCwg2cdegl_0@shg2CeleMz-1A6g2cg.za.pbo9ccttfas_DaeFzoc.e

-

Ha

(154)

G ^servo

²

^soluz

^-

stationaries

ylH=e%2 ^eylttetrz

^.

Porche consderaino polo _yo > e- kk

,

G _saw _zotsi

Det"Ey

^. ^<

^eh

ID _yo ^> ^eeao

(155)

Case etkcy

^. ^<

^etrz

→

yet ⁾ ^e ( ^Eta

_,

^etvz )

the _Amin ,Tmax )

⇒ Tmim=

^- ^a _,

Tmax

_, to

⇒ log4yiH ⁾

^<

₂ ^I

ytttlogclogtyctl _→ ⇒ _Ye

^'

_decrement ^)te{ ⁾

^<

⁰

.

(156)

Per _mwnotowa

F ftp.yitt.L

^.

^E ⁽ ^yo ,e£]

→ F ^Gm _t→ y

^'

^A)

⁼

log ⁽ eights

- A

2 ⇒ per Mteor

^.

esmto ^to srdereavere

log ⁽ ^log

^?

^at that

^.

^OH

L

^- ⁼

e%2

_s

₍

.

=

e-

'

K

(157)

L ^' ^uniee possibilities ^L

^- ⁼

ethos

^.

Aualofemente

_,

sr rode ehe

Flim

t

→ to

Ya ⁼ ^e-

he

(158)

Convessta _/ ^Con ^can't

y

^"

Hkdft y4H=dz ( _log (log4yHkD

=

YYH

.

2log(

g.

loofcycttt

^'

E) _ylt )

^.

Perdu ^'

eogzcy ^He g) ^{> 0}

^e

^944>0

^,

R seguo ^oh

^'

^y

^"

^drpeude ^solo ^dial

seguo ^di ^y ^'

^e

^di log ⁽ ^y ⁾

(159)

Rreordoehe y4HeO

→ sign ⁽ y

^"

)=

^-

srgncwgcy ⁾ ⁾

⇒

yaa ^,

[ ^EO ^se ^se ^•8Ifmy÷I logcyttllzo

^.

cos yutzs

(160)

01

^:

L

- =

etro

> s

{ ^L ⁺

⁼

^e- ^" ^a

^<

^s

e y shut

^.

deerescente

dwndi _F _! t* ^ER ^tale ^ohe

y ⁽ ^t* )

⁼

¹

,

es ^'

he che

.

per tcta

,

YLH ^> ^I

^⇒

y ^con ^cava

.

put ^> ^t*

_,

yutce ^⇒ yconvesso

(161)

Case yo ^> e%s

dwndi ya ^> ekra lttecmrm ,Tma× )

.

Si redeem

glut ^>

^o

^Ktectnwm ^{.Tma× )}

→ yetresante

^.

_C*)

Quimdi e%hYtHsyo ^the ⁽ Tmm ,o]

^.

De puerto aoneludoeho Town

⁼ ^- a

(162)

( _appbo ^Meteor

^.

^sul prohngamuut

delle slide ¹³²

_,

offure ^ossowo ^eke

hocresuta _snbemeene Amped

solution

,

astute

a

( Tmim

_, ⁰

] )

^.

←

a dunno share ehetmax.tw

Uewfreo encore die ho

,

Weser he sublimate Angola ^solution

_,

Nstuue

_a

To Imax )

^.

^Come ?

(163)

.

non hu torn

^one

^stems

^a

priori

^sw

ya

^come

^invea

ⁱⁿ

^C* )

^-

Den bare lefopietoidi log ^!

De ^Ricardo ehelogcyley ^Ky ^>

^o

A)

12 ^Porche ^' _ybgm

,→

boyd

Hea

^,

Vale amehe G)

Far ^B ^so

^:

tyecates ⁾ ^bogfcyl ^s Ayt ^B

(164)

Qwndi sr he

,

Angola ^solution

ya

,

fct

^,

^y ^HH _log ⁽ loafcycthtes

PEN ^eogeqycts ⁾ ^-4£

Puget ^Ay ^He ^Bttlz

Pored _per _yo _> ^eh

^a

_his _ohe

(165)

(166)

y ^{' '} ttf Hiya ^> ⁰ ^e _you ^>

^o

,

hoanchehsdrsnguaglranaeconimwdneiIfctryLtH1eAlylA1tBtz10Kcresertasublrneaw@mfEesduhonel.asTmax.t

is

(167)

Con ^lo stesso onto ^di fume

^sr

^neale

che

y

^'^'

^CH ^>

^o

^Voter

→ y ^I ^comesse

^.

£ R ten

^.

^dell sin

^' ^a

to to

sr

rede die

L

^- ⁼

line

t

_→ _→

YCH

⁼

e%a

L ₊

=

Gm

t

_→ _to

Y ⁽ ^H=

⁺ ^•

( si ⁱⁿ megionov ^Puerto per ^ease

Assuruso )

(168)

Il teorema del confronto

Siano u e v le soluzioni di ( u ⁰ = f (t, u(t)),

u(t 0 ) = y 0 ,

( v ⁰ = g (t, v (t)), v (t 0 ) = y 0 ,

con f e g soddisfacenti le ipotesi del teorema di esistenza e unicit`a locale.

Supponiamo che

g (t, y ) f (t, y ) 8 t 2 I , y 2 R.

Allora

v (t) u(t) 8 t 2 I , t t 0 , v (t)  u(t) 8 t 2 I , t  t 0 .

(169)

Es. 15.

Dimostrare che il dominio massimale di definizione di ( y ⁰ = y ² + t ² ,

y (0) = 0 non `e tutto R.

TES

^.

^Anna ]

f ⁽ try |=y4t2 ^e ^e

^•

CRY

.

Oh Z ! locale

dom ( ft RZ

^. _,

sivseravohcale

^-

^ok

his fchy.tt?ey2NNhiecresartow

(170)

snblrmeare !

Won _peso appbeere ^R ^ten

^.

^F ^!

globale

^.

y ^I ^def

^.

^su ^Ctnwm

^,

^{Imax )}

^.

←

mwstno

ehey ^OBBPARL

^,

^woe

^'

^due

ZCH=

^-

ytt ⁾ ^concrete ^con _y ^Ct ⁾

^.

Quest _seguue

^'

dalfalto

^die ^Z

^risdveeo

Jesse problem

^a

^di Cauchy

^di

y

^.

(171)

Tnfalti

Z '

It )

⁼

dgt fyft ⁾ ⁾⁼ ^t

^I

⁾

^.

^y ^'t

^- ^t

⁾ ^a)

=

y

^'

^Eth ( _y ftp.t ttf

=

(

^-

yft ⁾ 12

⁺

^£2

=

ZZ

( ^t

)

⁺

t2

e

Z lot

^-

y ( ⁰¹⁼⁰

^.

÷

no

striemo

pwnudi ^che ^Truax

^<

^to

Pee Wspauta

^avumo ^anehe

Town ^>

^. ^- ^a

(172)

Pwadrmentot

^.

Per asswolo Tmax

⁼ ^to

.

Usemolo R ten

.

dell

^'

esntoto

^sr

^olimo 5mg

che Lt

⁼

^Am

qt

→ to

Y 't )

⁼ ⁺ ^a

che esrste pocket yeiorescente

^,

in falti _y

^'

⁴⁾ ztzzo ^tt ^Eto

^, ^to

)

Graeme yeoto

^e

^L+=ta (

^e

y

^e

'

continue

_,

^F ^t* ^>

⁰

^tale ^die

yk*l= ¹

(173)

Confront laseluzoneylt ⁾

^,

^die ^wsdve yteeuz

|y(t*\=z ^Con ^he ^solution ^del ^pb

^.

i.

=gkiz ⁾

di eououy { ^t

'

22

'

.

Pure ^ten ^del

Zctats

Confront

_,

^srccome

f( _try ^)=t4y2> gctiy )=y2

^,

^dedueo

eke yk ⁾ ^> ^zct ⁾ ^ttzta

(174)

Posso _vsolrere esplrcitomenle

{ Êttt ZCt*1=1 Êat ⁽ Ê

^A ^var ^. ^Semrau

^! ⁾

e two _{2- cH=}

1-

t*+s

^-

^E Qwndr ^ho

yHzzu→= ^1- ^4tzt*

t*ttt

ha ^line ZCH=

+ A

, pwnobaeneluolo

t→H*tsT

(175)

die line

-

ylttt

^is ^.

^Assundo

_,

ⁱⁿ

t→Ct*tr )

wutneddrzione

con

oe falto

^die

^Imax

^=t ^is ^.

weeolimento2_puassmdolmox.tn ÷

^.

Dall

^'

eefwezinedrfferenzrale ^dedueoche

g. Tie

^-

¹

^-

^he iiuegno

^su

ogm

^'

nlmteurallo dllbe

found ^Iris ^]

^,

^tszs

(176)

otteug

^,

^tszs

s

^-

i. fsssdt

⁼

µ ^Y¥ ^at

4

^?

^Alt ^E

given

^.

^fstjttysdt %

42++2>-72

=

fsoftfff ^, ^)dt

staff ^,

^-

^gets ^,

(177)

029

_,

^setmax.eu

_,

sr he

bms

→ to

Ylsttn ^come groiossewab

^.

Quinon MO

GEED ^.GL#tfm ⁾

+

is

Aviv alla couineddr none

+

A

-

1 ya ^,

^-

Assmdo _!

(178)

(179)

(180)

(181)

(182)

(183)

(184)

(185)

(186)

Es. 16.

Studiare (

y ⁰ = ye ^y

³

y (0) = ↵ al variare di ↵ 2 R.

-

(187)

(188)

Studi qualitativi per problemi di Cauchy