Eserzi Cauchy con metodo di somiglianza e di Laplace - parte 2

Y

"E-1tce

y A t

-1

Ce

+

= + .

y(0) 1

= 42 0

0 1

+

+ =

y(0) 0

= 2

1( 0

+ 4 %

+ 0

- =

- -

4 4 1

[n

C1 1 =

= 1

C 22

12 0

=

+ =

- Yet te

t At 2

e - 1te

y +

+

= Ge-ht

4 g" 16y =

- 1y)(0)1

y(0) = 09

p 6 164

16

16 24

=

=

-

- = quindit

1

1 coincide

Des

4t Cekt

-

(1 2 +

yo = 4t

-

Ate

yp = 4t

A2-"t-4ste

yp = ↳t

-4Ae-hE GAe- -

y

p I6Ate

+

= _ Atet be-ht

**

De-hopte 16 =

- -

16At-16Dt 4

8A + =

-

E 81 1

4 A

- = -

= 0

0

160

160 = =

+ - 4t

-

yo 1te

= - Cre Ce"-ste-t

y. +

= -4t

4t "

t

↳

- -

y Ge zte

Le

4Ge

+ +

= - -

y(0) 1

= *

Cr2 0

2 1

+ =

-

y(0) 1

= 4(2*

4 22 0 1

+ =

- + -

4 C

(1 Cz

Cz

+ 1 1

= -

= q

4( 1

y 4

4(2

4( b(

1 + + =

-

+ = -

-

-

4 Cz

Ce 1

= - 1 4

C2

484

8( 1 2 +

+ =

=

= 6

Cr 5

H

1 =

= - 1

St Eteht

t

↳

-

E e 1 e

y +

= -

16 16 t

E -

y" 2y e

2y

+ =

+

y(0) 1y(0) 0

= =

2(f(t)) f(x)

=

2[f'(t)) ef(x) f(0)

-

= f(()

f(0)

f(x)

2(f"(t) -

1 -

=

it l

= -

1

S + f(0)

2ef(s)

(0)

8

(s) f(0) 2

+

r - - -

2 f(x)

+ = 1 1

2xf(x) 2f(x)

if(x) 2 + =

+

1 -

- d

+

s

f(x)(i #

2) 5

+ 2

+ 2S + =

f(b)(2 A

2) s + 25 + 2

+

28

+ + : S 1

+

f(c)(s 2) 3

+

2)

+ + = 1

s +

23sts

f(s) = 2

+ 213sts

# +

stE =

2

S

+1

S +

2

As 3

2AS S +

36

BC C

Cs

+ +

2A BS

+ +

+ + +

=

E A

B 1

A B 1 -

=

+ =

B + 3

20 c =

+ 20

3

3

20 C

+ c -

= =

20 3

A 3

2A =

1 +

+ -

- 1 1

A

A =

= -

- B 0

= 1

C 3 - 2 c =

=

l 1

# --

51 82 + 2

28

+ ↓

et completement

quadrett(ad)

del

i (A B) B

2

2

+ + A 2 DB

+ +

+

=

[a 64

=

2AB 25

= E S

A

A =

S

= 1

B

2SB 2S =

= B2

A 2AB

+ + 12

2 2 AB

+ +

5 1 +

1

+ 2AB 2

+ -

-

1)

(s 1

+

F 1

+ -

-

8 + 2

28

+

1

t

- -

E + 1) 1

(s +

+

ebtsin(at)

Raz = e[sin(t)

t

e- +

& t

2

y" y 2y

+

+ =

1y(0)

y(0) 0

=

= f(s)

2(f(t) = bf(s) f(0)

2(f'(t)) -

= - f(0) f'(0)

12 f(s)

2(f "(t)] -

-

=

e-t 1

= 1

S + f(0)

vf(s)

f()

sf(0) -

2f(s) +

-

- 2f(s) 1

+ = Ste

-2f(s) ef() =

2f(x)

1

1 +

- +

-

f(x)(x 1

2) Ste

S +

+

+ = l

S +

2)

(b

f(s) 1

S S

+ + +

= +

1

5 +

8(x = )

+=

# St

Ste

S 1

+ 2 S

As 21

As BS C

CS

BS +

+ +

+ + 25

+

= 2

+

+

E B

A 1 A

1

B

+ = -

=

A B 2

+ C

+ = 2A

2

c -

20 2 =

+

C = (2

A)

(1 2B)

A 2

=

+ + -

-

A 1 A 2

+ 2 2

+

- =

- &=

4

1

2A =

- -

1/

B 1

= -

1 1

2

2 c =

-

= 12 +s

- 2

+

S 1

+ B2

B) 2 a

(A 2aB +

+

=

+

g2 A S

A =

= BE =

1

2AB B

S

2SB

S

= =

= -

8 Y

4

3

3 +

3

- +

2

+ +

2

+ -

(s 1) - Y

+ 2

+

4) 7/

(s2 +

+ ebtsin(at)

Raz = -ebtw(at)

e-btcort b

e

et52 *

= ta

ept

-K t

E = e i

i t

+tor

+

E .

+

+2

S 1

+ Y

t Yet - sin(t)

- ecor(t)

-

1 Ih

L -

+

2 en Trn (t

is >

- 7 "

veYr win(i t

+

Eet -"T n(t)

/Ext e

+

e

+ n (

"

set e-4t cos(t) e

+ e

lette t (FF

(t)

Scor vin

E 2e3t

y" zy zy =

-

- 3t

1y(0)

g(0) =

0

=

= 2h 3

8 -

2[f(t)) f(s)

= bf(s) f(0)

2(f'(t) -

= f

/)

-f (0)

12 f(x)

(f"(t)]

2 -

-

= f(o) 3 f(x)

2(ff(x)

xf(0) f(()

xf(x) =

- -

-

-

- 3

3f(x)

-f(x) 22f(x)

0 + 2 - =

-

- 3

f(s)(r 3) z 5 -

2) 2

+

- - = 3

5 - s 6

+

25

2 33

+ -

-

fls) =

asto -

-3)

-es 3

= -

se =

, 21 1

(s -

3)(s 1)

3)(s +

-

- "

* is to t

"

i)

s 3)2

C(s

3)(c 1) 1) s

A(s B(s 8

3) +

-

+ =

+ -

+ +

- 9-

52 65

+

As -3D S-SSt8

6Cs

3D 9 C

2 AS Cs

BS =

As +

3AS B + -

+

+ +

-

-

& A 1 A

C 1

+ c

= -

=

A S

3D 6C 6C S

B

+ 2A B

-

+ = + =

-

- -

-

-

9c 8

31 =

+

3

+

- 8 9C 3A

+

B -

=

8 9 C

2A 3

+ 3D 6 C =

+ -

- -

- A 13

15 -

=

- 15(1 A)

A 13

=

- -

-

A 15D

15 13

+

- = - 1g

q

160 B

2 =

= = =

16 Y

bc

1

2 =

-

= (5) 13 3

3- E e

8

B B

+

9 +

8 =

-

=

B .

-

=

& A = 8

1/2

B = Y

C = C

-3) -

( 1)

+

1/8 7/8

1/2

t

- t

- -

3

s (S-3)2

- 1

S + t

37

+ st -

7

· L

1 te

e + ↑

. j

.

(2

-

t be

+

+ e !

at

-Are

I

°

-

a)

(S + trin(t) =

1)

(s2 + 2

tc(t) 1

s

= -

1)2

(s2 +

-at

ne

I Z

°

-

Sta !

is

ht

E y -

16y 42

=

- 1y(0)

y(0) 1

=

= bt

- 4

Le e = -

4)

f

2[f(t)) f(s) +

= bf(s) f(0)

2(f'(t) -

= f

/)

-f(0)

12 f(x)

(f"(t)]

2 -

-

= En

f'(0)

sf(x) bf(0) 16f(x)

- - I

- A

16f())

xf(x) 1

1 -

- =

- -

( 4)

+

16)

f(s)(52 +

=

-

f(d)/s 16)

= 43 5 Sth

=

- 4

S +

(s)

f 53 -

: 6

( 4)

4)(3 +

-

A 5

+ 3

E +

+ =

4

4)

4)(s 4)

c(s

B(s 4) 2

A(s &

+

+

+ 3

=

- +

+

-

+ 2 8S

AStUD 16

2 +

3 +

hAS-4As-16D

AS +

As S

Cs

4B 8

16C +

8(s

BS

160 ES

4AS + +

4As +

+ +

+ =

-

- -

A 1

+C A 1 C

= -

= 8C

5

8C B

S

B + -

=

=

16A 16C 8

4B +

- =

- 8

16C 16

16 2 32

+ + =

+

-

- 44

64C = =

A

c = a

at

A E

1 ne

1

= - = I

16 16 Step 2

=

. B

5 8

B -

= 1

1

4 1

E + .

.

-

· 4)

( 16 S-4

+

16 te ent

n

- -1

2 e .

.

(6 st

7y

y"

4 22

12y =

+

-

g(0) = 1 Y'(0) e

=

4 7x 12

+

- E

=

48 -

- &

2

2 4

4t

it Ge

Cel +

Yo : Ateht

yp = GAtent

Neht

Y ↑

= st

GDe"F 4De"F

y"p IAte

+

= + thate2t)

*

7 (A2"

IAte3t

4De" 4De

+ -

+ Let

12 Ate

+ =

8A 16At-7D-28Dt 12Dt 2

+ =

+

E 280

16A 121

+ 0

0

- = 0

=

SA A 2

70 2 =

=

- test

2

yp = 2 test

Get Get +

+

y = Get Ce st ce

y stest

3

= + +

y(0) 1

= Cz 1

(1 + (n

= Cz

1

= -

y(0) = 0 4C2 +

3(1 2 0

+ =

4(n 0

2

+ =

34

3 +

- 6

[1

3

C -

= = st

setzte

ent

y(t) 6

= -

4t

E y" 7y' Ce

12y

+ =

-

y(0) 1 y'(0) 0

= =

* 7x 12

+

- 3

F l 1 -

48

31 T

=

2

, Get t Celt

yo = 24t

Art

yp .

= 2"* ht

hate

Ip A +

= Ate4t

4De"

4De"

Jo +

+

= t

*

-De"-c8Ate"

16Ate e2Ate

Det +

+

· - 2eht

-

\ 2 4

Y A

8D + A L

=

- = 0

286

160 o =

120

t 0

+ =

+ -

2te4t

Yp =

Gest <test

G2"

+ +

y = 4Ge"* Ce

Ge test

y 8

+ +

3 +

= (2

E (2)

y(0) & 1

1 +

= + =

° °

y() 3(2 3222 2

+ + 0

+ 0

=

= 0 4 6

C

3

S 1 +

Cz =

1

C C

= - = 3

C2

3(

3( C -3 =

3 + -

2

+ 0

=

- =

623t steat

524 +

y -

= 4t 4t

E y" 7y' 22

12y 1

+ =

- e = -

4

s

y(0) 1 y'(0) -

0

= = En

2[f(t)) f(s)

= bf(s) f(0)

2(f'(t) -

= f

/)

-f (0)

12 f(x)

(f"(t)]

2 -

-

= zef(s) 7f(0)

f(()

-f(0)

s2f(x) + +

- -

- 12f(x)

+ 2

= -

S 4

-

12f(x)

75f(x)

x2f(x) 7 2

+ +

u =

- - -

4

s -

12)

f(b)(2 - +

75 + = 2

- 75

45 28

s

2 +

+ - -

* -MS 30

-est +

f(x) 3 H

= &

f(s) -st ↳

:

3) se - 3

4)2(3

(s 3)

-

-

Br =

A 0 see

, ) 4)

4)(s 3)

A(s c(s

B(z 3) 15 30

+

+ =

- + -

- -

- 32 85

16

As-3As-38s +

As-h8 -

128

+

* CS2