Equazioni differenziali e relativi problemi di Cauchy - parte 2

O

)"(e33 33) 53d3

3x 3x + -

+

= C =

-

#

23)

(x(5x - 3 1 dy

= . ·

- *

23d3

-x. )

- 3 x

3x -

. -

< 23dy "

e3x) *

e3 d

+ =

= - [3/

[23x) e *.

-e =

-

. costtilo

costerte auche

una non

>

- poso

5x Ex x23x =

- -

.

= x)

ex)

xe

--He3x -

-

=

-

Y

LEz (43 13)

9 :

. confronto/somiglianze

1

E y" xex

y =

- Corchy/

2 ente

y(0) 0y(0) 0

= = f(x)

y

y"-y

associeta

diff 0 =

=

Eq omofener

· P(n) =

( 1) y'

quando monce

= 0

= funzione conservativa

ha una

si

81 2

=

=

2

, *

, Ce

q(60) c +

= Ax2e* BxeY

(Ax B) xex

y(6) +

=

+

= Axe* Be Y

2AxeY Bxe

Y'(GI) + +

+

= 2Axe"

2A2 +Be

Axe

2Axe Y

"(GI) +

Be Bxe

+

y + +

+

= *

(4) +

GAx2

2Ae* +

y" Bxex

Axe 2 Be

+ +

= *

-Axe Y Y

*

AyeY Bxe

2 Be xe

+

CHe GAxe -Bxe =

+

+ + - -

*

Y

GAxe

2Ae 2B2 x

+ =

+

- 4Ax

2B

2A + x

+ = SA

SAFA

E 4Ax X

=

2B

2A 0

+ = (x 4)xe

222"

Ge *

y(G1) + + -

= = -

1 L D

A

.

C I X

(x 5)xe

ae ce

. + -

+

FD .

(

C Cz

C

0 >

+ -

=

-

= (4x

5)ex *

j)xe

(4x

Ge Ge 4xex

y +

+ + -

= + -

-

2

C I

.

.

FD 1

2

G

0 +

= - -

↳ t

22 4 +

= [a

1

+ 22 -

-Ce -Ce-

& C G = = -

=

= +

21

22 -y

= x 1 x

+ 1g

= =

(4x 4)xe

be 1ex *

Y(PI) + +

-

= -

)

-(e- since

=

Risolvente

Nucleo

:

° MeToDo

2 (* *

yG Cze

+

= '' *

el

E comparte ·

(x

- s

e

x(3 x) costeute)

de

=

, x

- ex

e -

-3 e3

B eB

-

e

-

3 3

+

3 x 3 +

- x

x - x -

-

e e e

= e

- -

=

3

e3

3

y -

- 2

e + forzente

Termina 3

- in

3)

As/Y(x 3 +

x

- - zeB

e

- .

1) ege3 - Be

#b ex 3d3 ex e

e

-

. . .

*

e*) * ! 23

Fe

*

- dy

* y

- .

* *

zex- e

(ec3 e

*

-f)

gl

*.

-E 23 /

1 [fe 2.

1e" x - --

x =

!

[fxe2

* * 122

1x 1e -

-

- e*[5x2 ]

+

exe-exerte * + x

- -

= -

termine +

[x

noto 5ex

appere run x

-

>

-

(55 27)

Lez 10 :

- 22(1 5x2

y" by 20x)

5y + +

+ =

- . le del

soluzione

cerce

si caratteristic

x polimaio

P(x) P

46 5 =

+

-

= 2IF1 EK

2πs 21

- =

=

= 82

-ex X

Ce Ce

YG +

= i) x

(2

(2 i) x

+ -

C2e

Ce

yG +

= exeminx

Cre

Y corx +

= . ec2orx

ex 5x

h(x) + +

=

-

Termine

Forzoute aeX 2ae *

i) y

y, = >

- =

Lee-see-sae

1 ?

have * 22 x

y # =

= e2Y termine

+0y1 pimo

1

a I

> =

=

- forcaute

xe2

e /Grnx)

evinu

Cee " * *

(2 y3

+

YG +

corx + +

= . F

particolere

y

dell'inomogener 8

y 3y"

= =

ii) brinx

xe2(acorx +

Ye = 3

1 2 brinx) (-asinx

e(acox 2xe(acorx

y *

beinx) bcasx)

xe

= +

+ +

+ +

2e(acx e

e(-asinx box)

beinx)

y +

+

+

= /-avinx book)

brinx) exe

22(ac0x 4xe(acorx *

brinx)

2 + + + +

+ +

+

e) - /-acoxx-beinx)

*

X

brorx)

2x22x/-asinx

blosx)

3 asinx +

+ +

+ + b 12

=

a =

22(1 5x2

y" by 20x)

5y + +

+ =

- .

Gabrielization

e /-asinx /-acoxx-beinx)

Y

bcorx)

~

bloox) asinx x 2

+

+ + + -

) -asinx

* bcorx)

ze " /-acorx-brind

2X brinx)

xe(acex + xe

+ +

+

+

- e(1 (0x)

+

= boox)

2)

brinx)

x(a(2x x(- a coxx-brinx)

sinx

a +

+ + +

-

(1 200x)

+

=

zasinx-2bloxx 1 + Cax

=

-

iii) ax2 bx c

+

y3 +

= b

y 2ax +

= 2a

y = 5ax2

4b 5bx

-

8ax +

2a +

-

- - 1

E a =

& 5x 1

a

5ax =

>

-

= 8

b /5

=

56 0

8

36x +

d =

8ax -

=

+

- s

C =

33 D

5

4b 2a =

- +

0

Sc

2a + =

- 2

Ex

Y +

+

x

= COMPLESSA

VAMANTE p(x) gnn

P(x)eXY

L =

E coo(px)

y = esin(px)

P(x)

Ly ife(X)

f(x) fn(x) +

= = eY(cowX

y" 4y) 5y

+ =

- (2 i)

71 =

=

2

, A K

E

i)x

Axe(2 +

y =

= (2 i)x

He i)x +

+ i)Axe

(2

y + +

= ilx

Axe

i)A2k i) x +

i) y

e i)

+

+ (2

(2

(2 +

y i)Ae + +

+

+

= i)x

Ay(2

(2 i) x +

i)

( i)

+

x

+ i) (2

(2 Al

(2 +

i) Al + +

+

+ (2 i)X

(2 i)x 3

(2 i)

E +

X 1x

+

+ i

(2 +

i)Axe 5Axe

(2 C

A

4 e + +

+ -

>

- ~

-

- i) e(

i) 2A2( i)x

i) 4A2(2 ilx

(2 x +

+ +

Ax(2

+ + + - -

-

- -

-

- (2

(2 i)x

2(2

~ i)

i) * x +

+ +

4A(2 i)xe 5 A e

+ =

+

- -

-

- i)

2A(2 i) 4aX)2

i)

AX(2 4A Sa

+ + 1

+ +

+ =

-

-

* /

2Ai Axi

2Ax 4Axi

4A

+ A

8AX

+ 5

+ + A

-

- - =

-

6AX-3Axi SA

2Ai 1

+ +

- =

&

1

+

2Ai A 1

1 ;

A =

= = - 2

i)x

-xie( + i) "

y (2 /co iniu

-

= +

= = +

isinx)

%(200x

"

1xe2 +

= -

-Exe 2x

Icorx Exe six

+

.

= -

- fr(x)

if2(x) segliere lo siluppo reale

devo foriente

termine

poichi il e REALE

xsinx

Seh y =

. 41)

/39

LEz Problema couchy

11 :

. = -

coeff cost

Ordine

y °

y 2 ·

+ Coox omgenes

Mos

y()

y(0) 0

0 =

= + 0

Cox

Ty kπ

=

x +

P(x) 2 =

0

4

= 1 1

+ = >

- -

If i

b =

=

=

- x1x -2 x

Ce

yG C22

= + =

ix

ix -

Cae

t

yG C12

= siluppo

-Fue

%G 2 C2

corx sinx

+

= a

2009 min

s

(3 x)

k =

, cosg sinx-cosxrimly

=

↳ rimy

cos23

I e

coag vies +

w(3) =

NUCLEO um

RESOLVENTE 3

sin

- 1

k(b x) I I *

Nnxcorg Ning

coox

= -

, O & X

b x(3 f(3)dz

↳ X) 3

=

.

, Ginomogena

"Irinxconly-cooxsinly) To

! . de

ED 3

*

"sinxcog Adg-

# in s e

.

sinx)Ydl "sins a

-coox)

As *

feu/corgl)

31

* -corx

sinx . GI

Cox (u/corX) integral

#xsinx =

+ inomogenes

Cecoox C

+ sinx cooxlu/coox

YGI X sinx

= + +

SONZIONE DELINOMOGENEA

GENERALE e(casd)

! 'i

.

-i (1) 0

=

y(0)

CoFof 0

=

ED C1

O =

2

Col y(0)

. 0

=

Fi &

8 1 =

- /-sim

/Coox) +

le

C2 Coox-sinx

y C simx + 100x

X

+

sinx +

Cox

= +

- C1 0

=

D

22 = 22 0

=

Yp+ Coxxlu)corx)

Xsinx +

= if2(x)

Ly f(x) f(x) +

=

= [mf(x)

Ref(x) ix

ix fix

f(x)

C =

Cox = =

= x & =

ax"

ix i)

- (

axe

yy = -

+ ix

- iai-iae-iaxe"(i)

-

y" *

= 12 1

>

+ - -

ix -ix

ix-ia-axe

- C

y" =

- ia

= 9 1.

L

1 a

zia 1 Q =

=

= -

-

- -

- *

*

211 K

af

*

fixe

y =

= 1xi(crx isnx) 1x six

=

= -

ix

C

corx = & x

= f(x) =

Ex = fix

(44 44)

LEz 12 :

. in Cauchy

Ly f(x)

=

=

↑ 2y

y y

+

- 323

R

kx -

-

0g'(0)

y(0) 0

=

= - wolteplicita2

P(x) 28

x 1 0

+

= = n 2

x 1 =

>

-

=

1)

(x 1

+ n 2

=

0 =

>

= -

- - la

perche 2

n =

x

-

* *

(G0) Ca

Y C xe

IG0 e

= +

= a somiglicize

no

3e3

x(3 x) L

simile

diff

l -

=

, +2

x

e

ex xex

GREEN'S Tette

FUNCTION zeB

23

Nucle

Rissevente w(3) = B

geB

e3e3 2 0 X

-

+ (a

1

I I

*

3

3 3 x

+ x +

x

x)

x(3 3 *

xe-ze

e

e

xe X

- -

, =

= = e2g ei

Be

23 .

3023

+ - 3)x 3)

ex -

= -

"

6 +(3)dy

x)

k(3

y =

= , .

= Anzente

Tennine

&

(ex 3(x y) a

y -

+ = - y 2

+

( (x 3)

z -

= - "(-

3

et) e

y = = -

e) -

F

= -

y dy e

e +

= O -

5 dy e 2

E +2 -

+ di

y -

xe

=

= g +

& %" et

e-gindy dz =

91 3

xe" In + -

-

= 2

21] g] +

" eng

xe" en/3 22

+ +

g e

+ -

=

= 2 3).

e( 2)en(3

y +

= + - 0

2)

ex(( 2)(u(x ((0 +

x

2) +

g + -

+ -

=

= YI(x)

2) x]

en(

[(x

+ 2) =

e +

y -

.

=

= >

[xex

ex 2)

y )

+1 *

pen

+

= -

= [(x 2)e(2) ]

x

-

+ 2) -01

(( 2)

(x

+

1 1

+

+

. . -

Integrale

generale

yf(x)

(2xe"

Cre ausgementinomogene

+

+

yG = en() x]

[(x

(2xex

=(e + 2)

+ e + -

.

96 +

1

Col ce

. 1 0]-