Esercizi Cauchy con metodo di somiglianza e di Laplace - parte 1

<B B

4 =

= -

- A 2

1

c = =

A C

- +

t -

(s 2

3)

S 1 -

8 -

3

- I

2 1

t

- -

53 Fe 3)2

(S - It

t

ze3t

y(t) te

et

= - Carcty STANDEND

↳

E -

" Se

y 4y

+ = 1y(() 0

y(0) =

= - se

x +4 0

= DO

-

12i

= X

4 =

-

- C2sin(pt)]

247(((r(Bt) +

yo = 2 0 B 2

=

= C2ein[t]

[C1 wordt)

Yo(= e' + A

essendo grado

di

Ae-ht

Yp(t) = 4 t

-

yp(t) 4Ae

= - 4t

yp(t) -

16 A2

+

= 3

-

y" by Se

=

- -4t

4(Ae(

4t

- - Se

16Ae I

+ ht

4t ht -

-

- ser

Ae

Al E

16 +

- -

(A 5

16A =

+ 1

A

20A =

S

= A yp

Sosostriso IN

Ae-ht

Yo = ht

-

1e

yp = Yo YP

CASOMMA D

Sowmore

LA 4t

&H -

cor22t) te

g(t) C

(1 min +

+

= ht

-

2(2cos(2t)

2(1sin(2t) e

g(t)a + -

- -o

-ot =

12

*

Casin

woo

Ch + +

E

4 1C t

- 1

g(0) C1 -

+ =

= -

= Cz

y(0) 1

0

1 =

2 Cz

0 =

= - z

It

(2t)

1 Corkt) ↓

y(t) + +

vin

-

= TRASFORMATA

CAUCHy DI LAPLACE

E 52-4t

-

y -g

by

+ = 1y(() 0

y(0) =

= -

2(f(t)) TRASFORMATE

f(e)

=

2(f)(t) ef(z) f(0)

= -

2[f"(t) 12f(x) f'(0)

rf(0)

= - -

2[f(t)]

2[f"(t)] So

Stltrisco

- -4t

- All'equazione

[f(t))

f()

-f() Je

th ima

12f(x) =

- - P

a gia i

applichiano

- - ef'(d)

iniziali f(o)

valori

5e4t

4f(x)

04

if(x) otHerIAMO

= &

+

+

TASFORMATA DI

LA Jest gox

L

trovp

gsi =

- ·

as

replace L'Intespace

CON 4)

t(

Se-st-htdt s

Je -

Se -

- dt

- 3

L at

Se =

.

: *

Jent nt 1etf-r 3)

-

e = 4

-

- -

SeGUENT

OPPWN APPLICANDO L at L

at l

l

2

- e I

e -

I - A a

-

a

+

1

quire o 2/5e45 = th =

4f(x)

14

if() =

+

+ 4

+

A Gre

Si RACCO f(s)

=

4) Ruperto

f(s)(x2 e

+ Mesorro

SE Es

f(s) -

6

= 2

+h

4)

s(2 +

> s

- :

-

4)

4(5 + S

Alsh) - . (5th)

4)

4(s +

s

f(x) = 12th SOLIONI

cerco le

Numerato ne

De

2 S 20

8 +

+ - 20)

(s 48

- -

- COMPLICAT

80

+

4 +

-

- ↓

2 APPLIC

QUIND SeMPlci

Pr

- SEMPC

PRSAL f(s)

1 Bo + C =

--

+

2)

4(1 * 4

+

+ ↳

A t

-°

-

4) 4(1

* 2)(22

+h)

4(1 4

+

+ +

4L

4BS +

4(Bx c)(s 2)

4)

(62

+

+

+ us

-

20

+ -

- +

h) 4)(eth)

h)(32

4(s + e+

+ 162

4Bs2 16D) 82 hS

h(s + 20

4 + =

+ -

-

+ +

As - TROVO BeC

A A 1

+

=

E 3ft)

4B 1 1

+

A A 4B - 1

2 2

+

1 =

+ =

-

= -

=

- -

-

· 4(z B)

4 16B 46

4 16B 4

4B

+ (6B

+

+ -

= + + =

= -

-

⑳ 10

20B = - - 2

B =

4(-

20

4 16C 4B)

A z

16c

+ 20

= +

- =

o 20

16C

16B

↳ + =

-

- 16B

+ 16 24 +

=

Tsa) 3

= B =

c c

+ =

4

12 + l

!c

=

c =

= -

Flatt

Formula

C Del

Need

B e

A

Sostituisio , Seronc

Bo 12th

+ C 20

1 -

=

--

+ -

4(1

4) * 2)(22

+h)

4)A 4

+ +

+ per

e

Tessform permut TASFORO

de

Oa

fi

of 1

↳

E -

2

+

o th on corlat)

?

sin(at)

eatEA eat & al

o

A

/it

cost)

E 1 sin

-

& Sow zoNe

RECONDAR

DA LAPLACE

2(f(t)) f(e)

=

2(f)(t) ef(z) f(0)

= -

2[f"(t) 12f(x) f'(0)

rf(0)

= - -

on corlat)

?

sin(at)

eaty eat &

A o al

I

ze

I 2[etsint](s)

e 13

(s)

y ·

= ( 2) B

+

FDAL SeMOUC FRAH :

SEMPLIC I 5

5 +

a + = D

S

1 1

+

bs c

as +

t

=

1)

s(x 32 1

+

+ L

b d

as + c +

t -

- Denomngrope

s 3

25 +

+

52 2

25 +

+ 3

X(s) > 3

= I

-

(s 1)(8 2)

+

+ D

STANDARD

CAUCHY L

- L

B =

- 0

E 2cos(2t)

cor(2t

y" p 2

=

hy =

+ y (0) 0

y(0) =

0

= 2 y

y y

+ 0

=

+

72 A

x 1 0 3

+ + = = -

Vi

2 4 0 1 =

4

+ = = -

T

=2i

1

* Vi

4 4 E

=

= -

- =

= -

↓ IiB &

Cop

8) I 2i

e[(ncs(2t) sin(t)]

(

+

yo >

= dep

-

Bsinkt)]t

(Acorkt)

Yo DONO

CINCI

+

= /Acoodt) Bein(2t))

Bcor(2t)

(-2Asin(t) s

ot -

+ .

Yp 2

+

= 2Bcov(2H)

( - 212in(t)

4Bwn(t))t +

+

-GAcov(t)

(

y -

= 2Asin(2t) 2Bc(2t)

+

- 2Bc0(2t))

( 2Aein(t)

sin(t) +

-thB +

thAcor(t) -

- -

- Btsinkt)

4tAcor(t

2Bc0(2t)

2Asin(2t) +

+ +

- -

--

welt)

=

E A

0

IA-2D GA 0

0

= =

=

- -

1

2B FG

2B + 4B

+ = B

1 =

=

Bsinkt)]

(Acorkt) t

Yo +

= )

Atsin(zt

Yo = Atrin(t)

(2t)

Csin

Cecos(2A) +

+

y = It cor(t)

cordt)

(2t) (2

y +

(1 2

sin +

2

-

= &

y(0) t)

Strin

(2t)

Casin

Cecos(2A)

:0 +

+ It corkt)

cordt)

C11in(2t) (2

y(0) +

2

+

2

0 -

= &sin/d)

1

/0)

Cncos(0) (2 sin

+ · D

+

(2 ·

100

/) 2

(sin .

+

2

-

E (1 6

=

2( 0

= Atcorkt)

y =

E coace

Cauchy

coolt)

"

y hy =

+ y (0) 0

y(0) =

0

= =

cos/at)

2(f(t) f(x) =

= a

+

f(0)

sf(x)

2[f'(t) -

= f'(0)

ef(0)

f(x)

"(t))

(f <

2 c -

-

= 4(f(x) =

f((0)

1f(0) +

12f(x) =

-

- e 4

+

12f(x) 4f(x) I

+ = -

82 4

+

4)

f(x)(2 [trinkE)

-

+ 2

= = <

4

e + [teindt]

2

f(x) =,

= n

p

4 trin

1

=

42

(5 + 1 B

A +

>

-

-

(* 3)

1)(x 1

- x

X

+ + 3

-

1 -A B

+

Tte) -

1)

(X

- -

T A + C

↓

-

1)(x

(x 1)3

X 1

- - -

1 1

-

- =

- 1)

1)(x

e (x

x + -

Ex A

: A

As = + 12

(x +

1

x

+

2

(( A B

+

x +

-

) = - 1

+

x 1

x +

&

AtB =

1- =

1)2

-

1)(x

(x2 +

+ C

** B +

1)(x B

(x +

4)

5x +

+ 1)

+ (x 4)(x +

& +

A e

= xz)

x1)(x

a(x

4

1 1

41 -

- -

= -

,

2

a ,

= d

# +A S

Cs + I

-

+ e 4

+ 4)(s 2)

d(s

4)(2rn)

4) B(x -

A(s 2)(5 Cs + =

+

+ + -

+ +

- 4d

dS

4B

BS + 4

c +

+

4) +

(A/

As +

- BS" hcs2-16Cs thas

4252 ds-hds

3

16 B (s + +

+ - Abd

As 16 A -

- Cs" ds-ndsthds

4 48 16d

BS" 16D +

16Cs

+

16A

As S

-

+ =

+ + -

- -

& A B

+ 0 A

+ B

c +

= 0

=

+

4 C C + D

D

+ 0

0 y

=

=

- A +

B

= - 6

162 4D (D 6C

1 -

-

+ 1

- = c

- = =

16D

16A 163 0

+ =

- - 46)

16) +(6B

+ 0

=

-

-

ts

-He

A +

= 0

16B 16B

1

+ =

+

- 1

Afr 32B 1 B =

= 34

-

+ is

is i - n

& y zeht

7y 12y

+ =

-

y(0) 1y(0) 0

= =

77 +

x 12 0

=

- 4

K

8 =

- : L

est

Ce

yo +

= e4t

Atebt perche 4

41 =

Yo =

= t

<

Deh hAte

yo +

= 16Ate * haeht

GAcht

y +

+

= "t

16AtePt-7AC"t-28Ate

A2 + 224t

12Ate

+ = 12Dt

7 D-28At 2

+ =

16At

A -

+ D 2

2

8A =

7 D =

- 0

0 =

Steht

fp = teht

zt

Ce C22 2

+

y +

= pte

4

It

4Ge 22

y +

34e +

+

= 1- Cz

(n

y(0) 1 ( 1

(1 + = =

= 3( 2

+ 0

4(n =

+

yo) e

= 3(2 2

+ ze

4

4 +

- 3

6

(2 Le =

=

-6 -

(2 =

- steht

6237

52 +

+

y -

=

& y zeht

7y 12y

+ =

-

y(0) 1y(0) 0

= = =

2f())

xf(x) 7 +

zef(x) +

- - 4

s -

12)

f(s)(s 7

z S

+

75 -

+ =

- 4

s -

↓ S 28

75

h

2 + +

-

-

= 4 113 30

+

-

[s

2

1(s ste

a

=, C

1 2

B +

+ 30

115 +

-

- --

--- D

2)

(S-

S- 3 4

S - c(s 3)

3) 5

2)(2

B(c 15

3) + 30

a(s + -

=

- -

+ -

- 85 BS

s 16 GB

+ -

- 2

As 4B<

BC

8Ds Cs

+

16D (2B 3C

3BS

+ +

- -

-

+ =

- 15 30

+

-

=

↑ 1

A + B = 1

A B

= -

OD 7 B + C 11

=

- -

-

+ 16D 12B

+ 30

3C =

-

7 B

8 8 B 1

C

+

+ =

- -

- B

B 3

C 3

+ = -

= - -

3B

9 30

12B +

16 16B =

+

+

- B 5

5

B = -

=

- 2

c =

S

3 +

c = - 6

A

A =

1 3

+

= Ente

-

S 3

- eteht

sel

g(t)-be't +

- t

E -

" 2y) 2y e

y +<