Eraciones

en
difsencia a de orden

*+ bX +CX7=0
hamogenta que
0
+

2 +1 por
+
=
+

x at

e aftat batta e
=
-

xf 1
+ =

X +2
+

=
-
30 Mr, Me

Pavar a
=

-
b vb2
= -
4ac
b 0m, mf
=

2
1) X
=
Am, m,
A so vaicer
2X y (A By)(m compleen

↑
+

(a,b)
z
3/X (Acar + sent b
a
+

= +

x
=
+

r
r =
a be +

r 10
=

-
a
CONO =

,
8 (0,52)-

25

m b yb2
2
= 4c I 4 c b2:
= -

b
2
r2
2 =
=

31:xfast*-85a-de-6, .
de e
(m-2//m-4) 0
=

cavo1Tambar poutivar >

of
X y A27 =

B4t
+
d
mn 2 me =
=

estable infinita
es
porque
-
no es va a

G2:xy z
+
-
8x++1 16x1
+

0
=

->
my 4
=

m2 (m 4)(m 4) 0
=

-
8 m 16 0
+
=
- - -

-
4
m2 =

x = -
A4 B
+
yt
+

-multiplico
porque My Mc =
(3:X1 z
+
+

2xf +
1 3x)
+

0
=

2 4 12 21 8 2188
m2 2m 3 0 m
= -
- - =

-
+
+ = -
=

2 2 2

2122)
=

=
- 1 2
+

-
cavo 3

E mad
R a D

N r
3
r 53
=
cort=
t
2
co)
D 8 =

para encantar o toon

a 1 inverti

X- (3) / correosif((+)-Bunfat(15(f)
=

G4:2f =

cyf -
1
-Y =0 +
+
k + -
kf -
1

ky 041
=

- 1

Y =0, +
04 + -
04 + z
+ 1
-

1
-
-

Y Y f
- -
12 0 +
0Y
+ + -

x 0
=

Y+ ((c 0/yf+1 04=0

+

2
+
-

una dow a law preador para trabajar can

portivar
prindar
 1

3x
m 1
-
1
=

xf 2x+=0 -
-
+

y
+ +

+1 1 +

m3 -

3m 3
2f
2
+
=

Me =
-
2 -
( -

m3 -

3m 2
+
= (m-r/rm m
+ -

2)

=(m -

1)(m 2)(m +
-

1)
mi -

3m +2
(m
= -12(m 2)
+

m - 1

8
X
+y 2x xy ↓ 1
diverge
+ =
+

+
+
1

mi 1)(m m2
1 0 (m 1)
=

2m m
+
+ = -
+
- +

m2)
↑ mS
2 1
S & & -

m
+
+

1 1 1 x3 - x2 +
x +

1 frx)
=

-
-
11 - -

1 - 1 1 1 & f((x) 3x2

=
-
2x 1
+

E
is
n =
=

M, 1m2 0.54360 m, 0.77184

= =
-

1.11514my 0.77181
- -

1.154
=

++
+z
-

8x +
+

1 yx
+

+ +2
=
->
Xi (a, af
=
+

a, fet
+

m2
Gm 1 0
/factorial
-
+ =

↑8m2 -
6m 8
+
0
=

61:
m
6
=
x4.8.48
1 6
=
1500
= =

&6 56

2 a
=
bi
+

Va
cott
t =
b2
+
O =

raota,
zacfteryefty (aptando tan en fetores las tan trate-e
2
I 1
=

4 =

+
1t x
+ +
=

1 0 =

k 0 =
 xy z
+
3xf
+

1
+
-

4X) f
=
- 1 -

X b0
= bit
+

x
- x8 x8
=

+
+y A,
=
4) Az +

x + +
3x +1
-

4xf = &
z
+ +

m2 3m + -
4 0 = mn - =

(m 4((m +
-

1) 0
=
Mz 0
=

bo+brf abert bet 4/bobef= -

f -
1

parciale
fracciones

x +z+
3xf
+

1
+
-

4X) +
=

2x8 (b
=
byf by2)
+ +

b0 by(+ 2) bz) z) 3(b0

+ +
+
+ +
b11f 1)
+
+

b) +
+ +

1/2
(b0 2) 2
-

br+ b2 1
=

+ +
-
- -

+2(b 4b)
3b 4b) (b 3b
-

+ -

+
+ +

Xf
 Juan Pablo Laledo 202123642

4f(x) -
= 12
Defnif(xxfry) -
f(xx +
1 -

fry
xy)
1 x2
+
-

por contradicion a
f(a)
·
= -

(xx r
+
-

xy) - X y
1 x2
+

1 (xx
+ (1
+
-

x/y)z ↑

-
2
-C
c - x
1 -
22 1 +

x2

-
c2(1 x2)x +

- x2 (1 c) +

-c-cxc -
x
- cx
2
- cc -
x

C >X

(xx (1 +
-

x)y)-
x

f(xx (1
+
-

x)y) min(f(x),fry))
=

f(x) =
-
x
( +
x 2)
frxx +

(r -

4)y) min
= (fas, fraryx (0,1) -

I
upaneman que Y

f(x) x2 yz
-
=

f(y) =
-

(1 x2)
y2
+

1 +

f(xx +
( -

xy)
-
=

(xx (
+ -

x/y)
1 + (xx (1
+
-

x)y)2
-rxx r- ery
e
+

1 (xx (
+
+
-

x/y)z
-
(x2 122411 -

41xy 11
+
-

y2yz)(1 y4x y2(n (x=2( +

-
+
-

xxy =

1 -

xr2yz)
-
((n yyx
+
-

yz(1 c)
+

c -

cy2c
-

yz -

cye
c)
y2
 2x.( x2)
+ -
x2/2x)
5f(x) x2 4f
-

= 2

x2
-

1
=

1 +
x2

r Pr =

Hertanxxy-faxadty.ax ( x
+

2)

He ) =

-
2X 19 I D
D
1 x22 I
4x2
B
+

=
2 -

-
2X 2x M x2
-
2 +

1 x22(1
+

x2)3
+

1 1B-
- 0

2X
1)
-

1 D D 2X
1 x22
-

1 x22
+

-
-
-
2X 2x 2 2X
2x 2
+

1 x22 H xyz
-

(1 x2)3
+ +

(1 x2)3
+

2X > & -2x +c - 0

1 x22+ (1 x2)3 +

2x > 8 -

2x +27 D ( x2)
+
- 0

2 2
x 1 1 x - 0
X > S
+

x = V x 23 -
1

=
x< 1:

R la
funcion is extrictamente
cualicancava cuando x >1
af +
2+
2

may x
y Sai x
2y2 y =

L
2

X(y 4x2
= x
yz +
zyz)
- -
+

10L =
2x -

8xx = 0 6x(y -
4x2 -

2y2) 0 =

dx
x?
2) y e

a
4
my
-

74
2
=

x
2y
+
4
=

3) xbh - x (x -

8xx) 0
=

dx

↑yo yray-xy e
=

tavo x F 0

074x 2(0)
81x
02xy 4(fy) 4
=
32x -
=
-
+
=

2 -
8H 0 =

4 x
2
= -

2 = 0X 2y -

y 0
=
x
2
1
=

1
x
z y = x

y 3 x 1
=
=

0
= =

R((1 (1,0)P2 =
f 1,0)
=
-

lavo y Fo

zy
↑ -

4xy 0
=

D 2x -
84X 0
=

7 ↑(0r 2y2 +

y
=

y(2
-

41) 0
=

2x -

0(2)x 0 =

2y
=4

2x -
4x 0 2
2 44 0 =

2
y
-
=
=

2 44 =
-

2x =
0
y
z
x
2
= 2X?
D
-
X0
XI
0
=

2) =10,5) 10, =
-

52)
 q
A
I
b) -
2

y sax 2y2
may x +
+ 4
=

I 8x 4
y
He =

8x 2x -
81x2y
-
4
by *D

↑yay
-

Dy2x
+ -
81x

Pn (1,0)= X 1 =

I 8 I
*

8 I 11 3
=

B, &
=

& & &

P2 1 N
-Ob
#
=
-
1 I

&
-
I I
132) 0
=

Bz =
-
8 & &

& I &

P3 10,5)
=
* =

I
-z

f
&
13) 0
=

3= 0 0 aie-E

↑z 222-re

Py 10, 52) x
=
-
=

& & -

By =
& ③ -25 2+)
+
0
=

472 -

252 z +

&

Pr a minima
local
ningun punto
es un masiva
 2 yz z
+
x
=
+
+

x(1 -
x -

y
-
z) m
1,n 3
=

r M +
1,...,h
=

1
3 & 1 1
r
2,3
=
=

1 2 I 0 frr

1
&2 I 1 -

132 f 1)(4)
=

- 0

1 & &
2 1 13-

( 1f 12)
=

-
-

- 0

2) van mining localle

2 x x/1 x-
y z2) n(1 x z)
y z y
= -
+
+
- + +
+ +
-

D & 2X
2y 2Z

B =

1
1
-

1
-

& O
n 3 =

D &
m 2
=
1 -

ax,
2X

&
-24 &
-1
27

& P -

24
-
1
27
 n 3 =
m 2
=

I & 2 & D

1
B
1
·
1 1
D 160
-

&
(3) f1)(16)
= -

1
=

-
=

& &
2 1 -
2

D -10 -2 & Pr (1,0,0,1,

= -
1) e mat

& -
1 I I
-

2 local

B & 2/3 43 413

-
· = & - -

& I 1 - 1 - 1 (- 12131 (1)(16)

= 1678
=

-
2/3 1 2 & I

p( 8,
, 3 1,5)e
1
-

-
413
-
&2 D -

1 I 2
/3
-

I
local
-

min
 b 1
=
-
(x - 12 -
242x(1 - x2 -

y)
18
2

=
- 2(x -
1 -
2xx 0
=

3/X(1 - x -

yz) 0
=

* =

= 2eBy
0
a) -
-

wxy =
4(x2 y2 +
1

lave X 0
=

aye
-

1 -

2(X -
1) 0 =
2 -
0
=

+1 -

x y =
0

X -
1 0 =

y ey =0 1 - 1 - 0-0

x 1 =

0 -0
0
=

Cavo *
= 0

①2 2-1 x
=
-

2ye x
Igualar:
e
=
2

2x 2
y
1 - X
=
-
23 * = -

2
x A X
1
-y 4
-
=
=
-

X 1
X =

-
-
1
3
2

y2
1 -
x -
0
=

Maler, volvian
hay
y/- y)
vari sin
no
1 e
=
 b 1
=
-
(x - 12 -
242x(1 - x2 -

y)
1d -
=
2(x -
1 -
2xx 0
=

4x(1 - x
2
-

yz) 0
=

dx
* =

2eBy
0

=
a) -
-

wxy =
5(x2 y2 +
1

3)bh = 1 -
x
-
yz 0
=

dX

lavo x 0 0
y
=

z(x 2xx (x 1) x (x 113

5
1)
- -

=
-
=
-

Reyy 2xy
=
23 y egg
en
23 =(x -
1)
y
=
em/ 2
y em =

31 -
x5 -

(m/xx4* e
=

favo x 0
=

y
0
=

24((1) 0
=

3 2 0
1 z(5 -

1) -

1 -
x =

1 x20x
=
=
= V 2-2 - 24/) 3
=

22 -

2 2x/X)
=

p(e,0) 0 4 0
=

22 4
=

0
2
-

-
=

pr- , 0 r

lave y
F0 x 0
=

3 x2 =
0 2 227 -
2 0
1
=

y
-
-

=
1 22 =2x
y

y
e
=
204 4
=
x 2
=

2
pro,i)
x 2
=

pro,-
L xy= x
+

y
+

x1(x
+
- x2 -

y2) x(1
+
-
x -

y)

4 1 2x1x x= 0 442( x -

y) 02
0
=
-
=
-
+
-

y
=

wbk z 5(x y2 = z
241y
+

0
-

x 1
- =
+

dy
04 6x y =
1
3x112 x y2)
+

-
=
0
-

favor R2 F0
y 4 0
=

I
4 1 x 0
2 X 1
+ -

2 3
=

y
- -
=

X 1 1 1 xz 0
y
-

y
+

=
- -
=

↓2 = 2 -

y
311(2 -
x2 -

y2) 0
=

Vample por Re=&

que

⑪y 2
2 1
- X 1
xz N
=

E
+
1 -
0 -
-

=
=

y 1
+
-

2
y
+ =
12 =
X =

E
2
1 0
2y
-
=

y =

1
favor x2 0 =

y rF0

2
-a -
x -

y2 e
=

1y 1
+ -
2x M 0
=

= 72 1
y 2 -
x2 X1 =
-

y -

141 -
=
2 -
-

2X 2X
z x2
y
-

x 1
+

x2 1
2x(x
=- 2
x2)
-
-

2 x 1 3
=

-
+

2
22 -
x
2X
x2)
21( x 1
=
+

1 tiene raien reale

11 x
=
+
· no

2
22 -
x
lavo D= D Xz 0
=

①y 1
+
0
=
② x 1 3
+
= ③
- 1 x - 1
y
=

Lave f0 x2 F 0

6
x+
yz z
1
y y
x 1 =

y
+
+
= =

v
2 y
=

y 1
24,4 42 (x 1
2x,y x2)
6y z
+ -

x
- - -
-
= =
=

y - x
(y - x
2x1)
+
0
=

21/(y
③(+(fz
1
+ -

x) 0
=

y x 3
-
=

y
= x contadicion
 Literar de ecuaciones

X 2Y
=

Xo Yo 1
= =

+1
+

y +1
+

1X-
=
llevar esto
a una lavación
en
diferencia
*
m2 -
1 0
=

X 24 1m 1mc
Yt
/ar+aza
=

+ 1 m =
1 =

1
+2
+
+ -
=

2/qx 1 X An Az( 1f remplaza

=
+

=
+
en t 0
=

x + 2 +
= xf 1 An
=

Az
+ 1
2(A
=
-

Az)
Y=
2x + 2 A1 An
-

1
+
=

* +2 + - x =
+
S

E
1xf y 1 -
Xf y z + 1rmef
Ar Az 1
1
-

- z +1
-

-
=

2
+

+ 1
+
= +

1
+
=
+
+ +

remplazo Yf+1
Az 2441
+1 xy zy f
An
= - +

fx
- =

+1 1
= -
z + -

z + +

1
- + f

12 ++

n
=

xy -

yy 2f
+

xf xy zf

⑳
z 4
=

2A1 3 1
-

+1
+ -

z
= +

+
=

An 3
= remplazo z
+ 1 +

Az 1/2
28 1)
=
-

+y x+x -

yf n
+

x1 4
+ +
-
371
2
= -
+

1xf 1
+
-
=
y - z
+

f
1
+

-dupyo z 1z f
=
-
=
x+ 1
+
-

y +1+

4y +1
+

xy
= -

zy f
+
↑

22 f =
-

Y 1
-
x +f
32 +
+ +

yy+2f
x
=

n y
+ -

las niodar valor is wman a las variabler

que dependen def

zf 1
+
=
-

x+ 2
+
-

y +
+

1
1
+

·-
xf z
+
-

21 1
+
-
=
y +
+

2
-
x +1
+
f
+
1
+

3/z++1 xf
=
-

y +
+2f
 ↑
Me s Ve
xf 2
+

- y +1
A
+

>42 -

Ve

2 +
+1
Vo
->
My -

x - 1 - 1

1
& 1 -

/A = 4
-

v 1
A xt
- -
-

-
=

1 P 1
x
- -

-1
-

- 1

-1 -
10

P(x) = 1 -

x)(12 -

1) (x 1) + - -
(1 -

1) (x
+

-
1)
=
-
(x -
1(x 1) +
+2(x -
1)

=- (4 - 1fx(x 1) 2) +
+

(x
=
-
11 -
x2 - x
2)
+

(x1
=
-

v1x2 x 2) +

-
-
= (x -
1)(x 2)(x +
-
1)

= -

(x -
112(x 2) +

Lix 1
=

X
y z

↑
-1 1 1
-
1 - 1 - 1 - -

A 4 =

~
-

1
0
-1 -
1
-

1 D I
1 O
1
-
-
-

Six =
-
2
=
A -
2I
1 & 1
3
-

2 - 1 - 1 I 3 -

1
1
-

1
S
1 e
-12
- - -12 -

D D &
- 1 -12 I 3 3

P
1
1 -2 1 -

1 2
-

~e
- 1
& d -

D I I
G2:y =0.404 =
- 1
7 0.68if
+

- 1 Ifproduccion

it 0.032) -
=

+
1 0.43if
+
- 1 if musian

y =
=
0.404 + 0.68i1 +
⑧ 0.68if y
=

+
+
1
-

0.40y1
+ 1

y 0.404 + 0.68(0.03247 1 0.43it

=

+
-
+
-
+
-

1)
1

y +2 0.40Y + 1 0.68(0.03247 0.43

=
=

+
+
+

y + 0.40y 0.68.0.03247 0.68.0.43if

+

2 =
+
+

1
+

3 +
+

2 0.40y+x+0.68.0.03247
= 0.43/yf
+

-
+

0.40yf)
3 +
+
2
-
y +
+
1(0.40 0.43)
+
-

y (0.68.0.032
+ -

0.43.0.40) 0
=

m2 -
0.82m +

0.18854 0 =
m =

0.82 = 0.521 -
4.0.18884

2
y An +Azxz
=
 En con matrien

0.40 0.68

A =

9.0320.43

1A -
x11 10.40=
-

1)(0.43 11 - -
0.032.0.68

Eraciones
en
diferencia
1) Polmamio caracteristico
Como caractertico:
of es armar el paliamio
Banco el mas
bendice
que aparece
pequeno
·

A Xs de bendice lo
er 1 m
er
pongo como
=

. Por cada que iba el bindice

unidad le aumento
s al

exponente
de m

0
G:
=

xf 2
+
-
x 3x+
+
-
1

más
pequeño
erf- 1

3x+ y(1)

3
-
-
1

Xf P un Pob caracteristica
X + 1x
f
m2 mi -

m 3/1) 0
+
=

* +
+

2
x m3

voluciones acuedo
parte
Solución de a
b) homogenta: is orman

la siguiente
tabla

Tipo de rair
Solución
Multiplicidad
a multiplicidad
1 Af M -
1)(m 3/4m
+
-
14m -
7) 0
=

real m 1
=
Mult
3

multiplia -of, trt, r

memulta
real, Mult
1
r
dad k tanto multiplicidad
region

a) Generan volución gensal parte namogenea a partir

de bancar

/X+(n G) baica 1)
=
Cz
+

2)
(bauca +....
 b) Solución
particular
if el
Minando lado derecho de la tabla, is
ecoge una solucion
de las
según reglar de la tabla de
carficienter
indetominadar
ensayo
ecuación
·i) Se remplaza es
mayo
en
original

G: (-) A
=
- 3 B2
+
=

c
+
en ecuacion Xxx 2 -
3x++ +

4xq
3 f
=
-

verice:

+ 1) ) +(1.3 + y 3 2

-(te
) . +
+

+
+ =
-
f
en
X +

+1 xf
*
f z:Donde ver la

cambia
letra en formula
nor
+ z
+

:j Igualando umyantes
calficienter a ambar ladar multa
un

interna para constante en

de ecuaciones dupar las algunas
funciona
gracion el truco de evaluar in valores partiar-
laver de f

4) Solución
general Xf (XX(n (xf(p
= +

A A
general namogenea particular

Ff Condiciones para hallar

iniciales constantes
que de
venian sale-

cion parte
homogener
e remplazar la volución general en los valores de
que dan

las iniciales
condiciones se revelve interna de para
ecuaciones
y
las constante
dupor

2
Rain
complejar en caracteristicar
palinamiar
Formula para
raiser
complear (no es tan
directo
el
analogo del
terna de ecuaciones difsencialer
Le = raier el polinamia
abien un par de
complyan en caac-

Asistica, entonces volucione carepondiente

baicas von

~ easrof, o enrof donde r= modular magnitud de

O =

argumenta angulo de
 Semplo raier complear:
a 1 =

m 1 =
5:tamo
= 1 f 3 I
↓ b b 3
=

a b

24
magnitud
r
(25) 2 N 2
=

12
= =
+ =

O El punto (1,55) o
cuadrante
argumento
=

esta
en 1

·Recorder
calculo modula
y argumento: (
Un numera
complyo
2 +b2
rab -t
= a

ptagoran
+

now
=

s
ra, bol

S
(
ein
Tan O O retan
=
=

ento cuadrante 1, I
en

.
O depende del
enadrante
de ra,b aretan/alfi in (a,b
enta in wadvante 2,3

b
O
arctan/*)
=
=

rangular conocido

Propiedad
de cuadratica

Le una am
cuadratica bm +c=0 no raice
tiene rabe

-b2- Yaco) y um raicen camplyan van de la

forma
a =
B: e pauible hallar o
y
O mediante
la guiente famula
r
,carO
=
= -

b 0
viendo c
(0,r(
permite
reto encontra o in revale la cadatica

Memotecnia conoceder March aprendere alguman uno, carenar

y tangente
O S 38 4
so 60 SD

/6 -u/y /3 fu/2

removeveo 2

e
3) coeficiente inditeinado

nolmania
de
grado.
0 A

18 At b
+

2A + 37
+
c
+
 Saracionen en
difsenda
E:Calentamiento
X +
+
z
+
2x +
+

1
x=
+
0(27)
=

o Hallow alucian general

iniciales haller volucion
b) Li condiciones
hubiera Xo=1 xs=1

a) Parte
hamagular X + +2 2xf
+

1
+
+
X
=
y0

·
m2 2m
+
1
+
0
=

para armar
polinancia
caractritico
la es:el
dinamica bindia
mar
baja
el avain can"s"por cada multiplica
unidad por m

(m 1/(m +
1)
+ 0
=
m = -
1 multiplicidad 2

H
Soluciones bania (-14, (- 1
Xn Anf- rf Acff- 18
= -

Lolución
particular
lado derecher (24
in habia
careción
Xp=A.25 paibler factor
de
mayor
repeticion
can
homogenea
parte
xf +
2
2x1x
+
x+
+

= 0(2)(remmaro en
original
A2* 2
2A2++ 12=erz) (donde no letaf ambie
par rubendic
+

+
+

Two by de exponente
pace factoria y
cancelar a f
A27.22 2A2.2 Aat erz

1
+ =
+

27
Xp 1.25
(Aa2 2A2 A) 012
+

a -
+ =
=

4A 4A
+
A
+ 0
=

A 0
=

A 1
=

Solución
general XX (x+(n (xf(p
=
+

X= G1 1
=
-
( 1 1++
+ 2 = -

b
misidle
Condiciones Xo=1, X=1/prmite
depuyer constante)
Xo 1
=

-an) 1 - +

62(0)( 1 2° +
=
1
R
6 1
+
1
=

2n 0
=

X + (
=
1f 27 +

X 1-2,f 1)
=

c2(1)1 1) 21
+
-
+
=
1

1 C2 = 1
-Cr -

C, =
-
S2

a) Prima la
organizar sancion

x +
+
3
- 5x +2
+
+

6x+ 1
+
5(24)
=
1
+

Parte
hamagenla X + +3
- 5X + +2 6x+
+

1
+ 0
=

m2 -
sm 6
+ = 0

(m -

2/(m -

3) 0
=
m 2
= m 3
=

An 2,3 c2.27
= +

Parte particular 5(2) 1

+

X A.27. +3 (u part
multiplica parte
repetir
-p para no
=

hamogenea
/el numa
porque el
palomia"of
x +

+3
- 5x +2
+
+

6x+ 1
+
5(24)
=
1
+

/remplazo X = 2

(A.24 + 3) + 3)
+
-

5/12+ +2+ 2)
+
3)
+ +

6/A2+5f 1)
+

3)
+

3127
=
1
+

En ya xnxer
-A B +

0
=

2A B 4 + =

Arx 2) B(x 1)
+ + + 4
=

cancelar A
o

y.
o tandro/dae valores a x
para

3153 63

Sie
24A 40A 12A 5
+
=
- + =

B
A S
Iz
=
=
-

A -
=

Lolución
particular xt
=
-
=

2247 1 +

Jantra/atra operan
(A.24 ++3) 3) 5/12+ +2+ +
-
2)
+
3)
+

6/A2+ +5f+1) 3)
+
+

3127)
=
1
+

emplaza en valores de f t 0,= f =

-

1, t -
= 2
 Solucion xf=fn +

(xp
Xt =
037 c227 + -

1 +

2+
+
1
d

X 1
=

23 c220 +
-

5+(0)(28 1 1
+

G 22
2 1 cz
1G 1 2
+ +
* +
= = -
=

23 222
72 2
1
Xn
-
+
+

1
=
=

34 22
+
-

2 1
=

32 2z +
=
-
1
*
3(1 -

c) 22
-
=
1

312
-

32 262 +
-
=
1 * 3/2 -

22 =
- 1

22 5/2
=

Cr =
-

X+ 237 12 g+2 1 mtiminar de f

+
-
-
= +

Es a

vaiee camplyan

Reecriba:Y f +2
-
44 + 1 +
-

134 +
1
=

Hamogenea
Y + 2 4Y 1
-

134 0m 4
=
t 42_ 4 (13
+
-

=
+ +
+

m2 - 4m -
13 =
I 2
m =
⑪ f -

36

raisen
campleyar: 2

36
- =
36 -
1 6
=

2 1 38
II-
m =

Formular para vaice complyar

m
si a
=
b:intancer
= de aluciane carespon
barrar

diente son r modula

= de m

con rot,
no unrof dande =rangular de m
 Recordar calcula de norma
y angula

E
r a
=
2
+
b2
artan
b
(andante
O =

actan() +ira,b)
cuadrante
en

2,3

M = 2 38
=
Lalucian bavia
A E /13/coufarst)+), (ne/sen/art((f)
A b

r 22 =

= 32 r
= =
13

O arcan =

(2)
En parte law bancar
homogenea varamar valucrane

Y +
+
2
-
4Y +
+
1
-

134 =
1

Men=c/wig,canfarstan).f) + de fuerten fart().

.
Lolución
particular
+) At+
=

(remplaza

(Art 2) B) +
+
-
4(A(t 1)
=
3)
+

13/A
+

+
+

B) f
=

exrficiente f:A -
YA 13A
+
11
=
A 1/10
=

inde:
casficiente -2B 10B 0 B 0
+ =

=
Er:
X X
+ 1
+
-
=
y+ -
z
+1
= +2
+
-
=

y +
+

1
-

z+ 1
+

y +
+

1
=

-
x + - z +f=

z xf 2f
+1
yy
+ -
= - +

X+ x +1 2xy 37 2
+
-

+2
+
+ -
=

me m 0
A 2 3f
z
4 (Af b)
+

+
- +
= + +
=
=

(m -

1((m 2) +
0
=

m 1
= m = -
2

arando adelanto
un

A,f 2)
toda
adelanta
Xn =
-

Ac
+

preodoracion *

92:X = +1 y=
=
x +1+

-
& 1 x -

y
-
1 & yf
y X +1
+

+1
=

=
+

(A -

x11 42 =
+
1 0
=

mr 11mz =
=
- i

12 =
-
1
-
A -
x1 =
1

x Ij
=

- 1 -
 1
↑
ix
y 0
x1
- +
=

ix
=
=

ix
y =

=-

(e-i
Hamagena victore

Xf
=of we caro-- we
unof-no we can of w,
umof
yf

+b2 a real
a
=

r =

b imaginaria
=
o que acompaña if
fr

D
⑦ CarO
I
=
O
cov
:

cuando no
hay particular
valuran la valucian
en to, of
 Juan Pablo Lalado 202123672 Luaver
Juliana 202123772

S
1)X=+ 1x 2
-
=

y +1+
=
xy
-

147 -
1

12 0 Xf -
2
Xf 1
+

-
I
yf -
1

y +
+
1
1
-

12

12 0
xf1x x)
4

x.(dt) (12
0
-
- -
= -

/A -
Nt) =

1
-

k -

42
2 14 2x
= - - + +

x,-
1 0
=

Xn 1 x =
-

1
Lava
=

1
=
I
(32) 3 8) 52e() x(i)v
- =

+A 21
=
=
- =

2
low D = -

1
17 11 (185 =
-

7-(8E 3(y) (9)m

= =

Xf

yf
=Anfef(i) Aaf
+ -

r))?

X1
35 e)()-(i)
a
=

sistema
de reacionen
y =
=

a
12a 2a 4
- =
= -

b a
=
-

X2b - 1b -
=

10/3
Xf

=
#
-
y =

a.fir-anfe) rerfin A
-
·
-

N
2 -1

A
Xf

It:
1X y 1x 21y + x 3y+ 1
=
-
-
-
+

-
-
=

*
-
1

#
X 4
y
zX+
f
-

=
t
=
=

-
-> 2

y= ( 4)
3 El
-

bagat
p(
y
8 4, 8). 3
-
-
=
3
-
-
=

en
y -
1
-
=
=
-
-

# ->
--
 E solución
X + 6x + yf La
-

1 y1
+
=

tie
=

a) X=

y inestable
+zy=
e
+1
+

= 5x +

X 647-
+1
=
-

x= =

yy
-

xq

DX1 5x7
=
-

yf

y +1+
-

y =
=
5xf +yf en
Ay + 5X
=
=

+ yf

(48
sx
=

X 0 =

4e

yy
-

=
-

3xf y 0
=

yf

E
x +1
X + 6xf y xyf 6xf
- -
+

1
+
= =
=

y +zyy15X+ y++1 2y=x

+1 5x
Gy 34
-
=
=
=

+1
= + -
+

1X++ 6x +1 14 +2+

5X
=
+1
+

2y++1
+

y
+

+1
-

x
=

2y +2+ 5/6x +
=
-

y ) xy+ +

1
X+
=
+

2 6x 5x+
+1 2yf
+
= + -
-

y +
+

x 30X +
=
-

Sy +zy +=
1
+

x
3 f +
x 6x+
=

1
+
-
5X - +
2/6x - + x + +

1)
X +2
+
6x
=

+1
+
-
5X - 12x+ 2x +
+

1 +
By +z
+

30/5y
=

+
+
1
-

2y+(+2y= +1

X +2 8xf+1 -
17xf
y
+

+2
6y
+

12yf ayf
= -
+
+1 +

1
+

xf+2 -
8x++1 -

17x) 0 =

y +2
+
8y +1
= +
-

12y=
m2 -
8m - 17 0
=

x
768 812) 41)
m 0
81- y +2 12yy
=

8y
0
= =
=
= =

+ -

+1
+
+

2
2

a 4b 1r
= =

16
= 1
+
r 27
=
m2 -
8m 12
+
0
=

(m -
6((m -
a
O aret 1
=

tan 0 1 =

d / m" 6 m2 2

XI
=

mzAcanlact 1+
=
un fact If
Xn A11617 Az(2)7
=
+

Forna facil
x +
y =6xy 1
-
+
f

h
vzAconlarct fact
X = 1)+ un
1
*
+1
*

viz
=

A
carlant?)
 *
x+ 3x++3
3x - y yy
E
+1
+

-
X +1+
=
+ =

+3
y
0xf 3y +
-

2
y+
-

A110 + ) +
n
+
=

x +1 = =

1
X +2+
3xf
=

+
1
-

y +1+
3
+

y= y
-
An)))- +3anot3aa 3 +
3
+

2 X + 3x + 10xf 2) 3
1
3yf
= +

2
- -
+ -
+

y+ 0
=
-
14 3(4)
+ 3
+

xf +
2 3xf
=
1
+
-
5xf 3y+
+
5
+

y =28 3y + 31 +

34f 2
+ 3x+
=
+1
- 0xf
+ 3/3x++3 -
x++
1) 5
+

xf 2
+

37 1
=
+
-

0x+ ex+
+
14
+ -

3xf 1 +

xf 14
2
y=
=

i
+

xf

* +z
+
0
=

m2 3
= =ÓÂàï -(14,31)
.

m 0
=

x- Arof =
↑

X-p Az1+ +3
+
↑

(14) 2+
+

A B 14

i
2
+ +
=

y = -
2

Arn(0) B
+
14
=

x
.

3 14 =

Az 0 =

no esta
a escala
Y
xy
=
0rett 17
+
xf p(1,0

Ar(14)
->

x= A1107
= + B
+
f

X=14
Es

f
·
XX =2x+ -

y +3
+

1yt 0xf 4yf w

-
-
=

->

DX+ 0
=

sy + 0
=
A

0 2x - y +3 0 0x +
4y + z f
=
+ = - - *
=

yf 2xy 3 y +
yx 1
-
+
= =

2xf
fx - z
3 3
y f 2(14)
+
= +
+

51 +

fx
=
-

2x y =31
+

E fx
=

X =
-1
 Taller S

I 3.4

E
X- =
ay=
3.4
+

-ayy xf 1 +

1
= +

5.4
y +1
= bxy
=
+

y f x
=
+1
+

f
3.4
a a

3.
1
3.4
Anf-Fabyte/as- te+
1X +2
+

ay
An(äb)
+ +
=

+1

y=
=

1 a
5.45) 3.7
+

2X +2 a(bxy
=
= + +

y 1 a
a54)
+

34
abxy
+

3xf
+

a
=
x

X abX as4f 347 1

+

+ 2 +
-
+

= +

m2 -
ab 0
=

/m-b) (m-Fabr =

m:ab mc = -
ab

Xn Anflab* Az f yabt
=

X +2 -
abxy 4f(as 3.4)
=
+

y(as 12)
=
+

x 04 =

+ 2 =
24 -
ab(y 4-(as 12) +

24/16 -

ab) 4f(as 12)

=
+

ab
2 =

4(as +

12) 2 as
=
-

*(16 -

ab) 16 -
ab

Xt
= Anlab* +

Az f yabt ↑ as-te

Anfb* *Anfrabter ↑ atre

*

X-
=

2k + +(a ak+ 1
-

+
+
(1 -

0k d =

0 d
m2 m(0/x 2) 1
+ -
+
- =

m =
-

b b2
=

-
4ac m -
=

10/c -

2)Irora -

22 -
4(n
-

2a 2

la 2 -
-
4(1 -

0) 0
 X +1 +
3x7
=
-
18yf

y +
+
n 2x7
=
-

0yf

=2)
"A
↑ ↑
3 -
x -
18
-
XI =

2 -- -

/A -
xe
(0
=

x)( - -

x) 36
+

x2 6x
+ -
27 36
+

x +64 0
+ 0
=

34
3) 04 3
- =
-

(
=

0
6x7 18y =
+
-

prapia
~Vector

Lix = -
3
x y 3yf
=

-*e ( =
wx- -

X= 3yf
=
6yy =

↑ 38
047 =

-3. Y
Un

⑤ Plantea
alucian

↑ 8 anfraf An/i) =
- rert

x (i)( ex Az(i)
=
- =

1
+
-

37

⑥ volución
verificar

18 12) -

/forten-(e/fort para es valaan e

que
time

ignahor
(i)(-er1 t
( 2/(i)(- of
=

=
foffer=()
rart

naturian
/fort=leftert de

litros arte
de tort
-

(i)(fr)( at
/(trart
referert refort=fffrort

z Plantea
volución
ka+x =k
+
rt
XE =
r p +

/A -

w(k 0
=

ra- were
en
-

Arte y por Alertr

12 (i) (a
-

691 188
36(1 32) 188 3
=
-

+ -
=

201 -
6P2 1
180=3
=

3 1892
+ -

201 1
=
62
+

01 1
=
32
+

Pr
1
=
Ecvaciones
en
diferencia
1 Polinamio
característico

af Como el arma un polinamio

caracteristico

Buco el bendice
más
·

pequeño
que aparece

A y de es ibindice
lo 1
=

NP
·
es
pongo como

for bendice aumentos al

exponente N
· cada unidad
que es bar el le de

Ej:X z
+
-
xf 3x7
+
-
n
0
=

· ub indice-s más
pequeño

3
3X+ - 1 13.(1)

Xf D x
13 x 3 0
=
+

X +1
=
- x

X +2
+ x 4

b) parte
solución
homogenea te construye can X v/
=

· Combinación barica
lineal

Xn Abaica
1) =
Az baissaad...
+

Mecanica varicas
valuciones

r 1,
multiplicidad rab -
rf

a multiplicidad, real s r trt, r...., f*- f

a) volución
particular
↓. e lado
taud derecho de la ecuación
il de
volución tabla
·
evoge una
ensayo rgun
·
vsificar
is se repite
con la
homogenea
parte es multiplica
una f

:i) se remplaza volución

envaya en
original
ecuación

Ej: (Xx) As B7 = +
C
parar
denacion Xxx - Sxxxxxx 37 th
=

remplaro
5/1.3+++(+ c) 4/137 37
/A.3+ 2( 2) 37+c) 72
+

a)
+
-
+ + + + = -

X X +1+
Xf
+z
+

si e
igualan carficienter
de terminar
umyante a ambar ladar
para de-

por constante recorder parciale

fracciones
o volución
general
Xx (x (n (x 10
=
+
+
-

4) condiciones iniciale

·
i
emplara en volución
general
valores de
que dan condiciones iniciater
2 Raie
complyar en
polinamio
caractertica

·
si m a
= bie un
par de raier complear entonces:

(xn A1(r ca )) +Az (rfsinrot)

=
+

dande:

r: modulo imagnitud"de m

⑦:argumento"angula"de m

Calcula modula
y del
argumento cuadratico
Propiedad
m a
= 1b: . no raicer
tiene realen

↓ ↓
real imagmaria
N
B
(b2 -

Yac0)
r a
=+b2 O r â
=

fan O b
= caro
-
=

2 ac

(art farto
fa, by
in I , IV
estar

⑦ , e
en rasar enter

Memotecnia angular
O 30° 4
so 609 ~

0 T/6 H/y 4/3 He

D 1 2 3 -
un O 2 2 2
2 2

I
car & Y 3. 1

2 2 a a
2

2
3
-
& 1
tan O 0
/ 3 2 1

3
Cificenter
indeterminadar tabla
mayor particular
volución

Lado derecho volución

encayo
.
Polinamio
grado n -Polinamio
general grado i

E: S E A
una constante

↑+2 E A +B
+

+2 - 1 E A 2
+

B +c
+
+

-3 +
y E A
 Lado derecho volución
encayo
·

Esponencial
rk exponencial
Mina por C

-Cr2Y) original
G:bra no necrito 3) (quito constante

Imgonametica Especión Acer (x+) 3

+
cau/xf
un (kf) ocar(kf)

A
G:6anraf (27) raff Iquito 6
-
ren B et
can
+

Esparencial
**

(Mima generabl
palinamita
esparencial). Mimo
.

por *

grado
nolmante n

(A 2
9:51234/17 c) (quito 5)
+1) *
+
BT
- +

·
o por
Esponencial ↳ (Mina
exponencial).(A u/k+) cav/kf))
+

enkf) a cowkf

G: 5(2)uf * ja (A
verot) Bcarr3f))
+ 5)
(quito

Polmamia
.

grado n
par ↳
(rol general) (newxf)(+ (pat gensall (car(xx)
unrkt) o carlk

5:(f 2)(un137)) + * (A+ B)(13) (At 3)(carlos

+ +
1

⑧.
.
Si a

Ypara
una volucion
repite de la parte
homogened la multiplica
por que se
diferencien

5:(y +(n cet

=
c2+
+
et

lado derecho Stef

=

tabla (y +( (A +3) e
·

ugun
+
=

A et
t
Bet =
Geeftcafet

2
(A + B)y2 0A 3e 3 y
- +
+ +

A
- 3e 3
+
2
y

ce cte +

Tip: cuando no
hay volución
particular
el
punto
de un intera

es (0,0
-Estabilidad en
ecuaciones diferencia
Una en
ecuación verahver las parte
diferencia
hamogenla
en estable
s ial

un volución
general tiende a dro

{
(r)
ei = 1
lim no =

1
1ir
=

-- ato
no existe
in
cualquir caso

5 Sistema
de ecuaciones en diferencia
:i Metodo
matricial

lantro la
· tema
en
forma

=A
E:vor
y +1
+
xeregee-e.
-
=
4x)+
yz

Truco 2x2
matriz

·
En 2x2 el caractictico
polinomio es M2-vara Al+(DetA1 0
=

Jara A Lumas
=
en la
diagonal

Regla baria:i sxer un valor propia de A

y
es un propia
vector

para
propio X, entonces (.
*er una volución
varica del

istema

9:= 22 15
112 x))
-
4

Truco: 42 -1 -

5/n +
y 0
=
A - xi =

x2 5x
+ y
+
0
=

(A -
xI) ((
=
-
2 -
x11 -

3 -

x)) -

2)
( 4)(X 1) +
+

0
=

6 2x
+
31 x2
+
+
-
2 0
=

x1 = -
44 = -
1 x2 51
+
+
4 0 =

N1 =
-
4

↑ -E ↑
8)-t 3-
-

=
Moratan en
ecuaciones
diferencia
-

Semplo 1:

xf+2 2x+ +
+x
+
- 0(2)
=

a Hallar solución

b) Xo 1X1 1 =
=

arx 2x1 xy 0(24) -ra+) (X+) crz

-
+
+

+z
=
+ +
=

C(2+ 2) 2(2+ 1) c(27) 0(27)

+ +

me 2m 1 0
=

= + +
+ +

(m 1)(m 1) + +
2(27.22) 2(27.2) +
c(27)
+
f(27)
=

m -
= 1
↑(2+) + y(2+) c(z+)
+
= 4(2)

(X /n Anf1 Az
+ =
+
1 -
17 4 +yc c
+

= 0

oc 0
=

X =
=
(X (n +

(x 10 =

c 1
=

X= Anf Acffet
=
+ at
+

(xxp =
at

b) Xo x n 1
depar An
=
=

permite yAz
Yo 1
=
-
Anf- 1 Azor- +
28
+
=

An 1
+
1
=

An 0
=

X1 1
=
D A
1 - 1 +
Az) -
1) +
i 1
=

Az +
2 1
=

Az 1
=

Gemplo 2:

X+ 5X + z
-
6x + +
5127)
+ 1
+

3
= +
+

Xo 1Xn
=
2
=

1) terminar #
organiza X-
=

xf+x 5x + + 6x f+1
5127)
=
1
+

2
- +

pab grado 0

Parte Corte particular

homogenla:
m2 -
Sm +
6 0
=

(+1 5124) = +
1 -
A124 +
+
3 D -(2+) 2 +

6/A .a 1) 527)
+

3((m 2) 0(A.2+ e+ e) 3) 5/A.2+ 2(f 2) 3) 3)

+
x 1
(m
+
= + + =
+
- -
-
+ +
+ +

Me 3 M2 2
lauf
=
=

- A S
(x (n A1134 Ac(2) A
S
=

2f +0A 12A s
=

24A
= - -

- +
+ =
+
=

ind: B -

5B 6B
+ =

1 - x 2B 1
= 3
1
=

xy An (34)
=

1,124)
+
-

f(24) 1 +
 A1(38 A2(28 g(20) 1 1
+
+ - =

Xo 1
=
*

1
5 1
Ar Az
+
-
+

An Az
+
-

3 1
=

A1
2 2 1
*
An
I Az
=
=
- -
=
-

Xn 3A1 2A
1 1
2
= -
+ - + 1
=

3A1 2A)
+
3
=

3 -

12) 2Az + 3
=

4 3A 2A) 31
2 3 A2Az
f
+

= -
- =
=

Gemplo 3:

y +z 4y +1
=
=
+
-

13y++f

f
y +
+

z
-

4y +1 +
+

13yy =

Pate
homogene (X-(n=All is" can fact (f)) An (es un rant(+)(
+

m2 -
1m 13 0
+ =

=
m -
=
1 -
4) 16 -
52

m 41 36 M
4
16:
1 3:
= -
=
m2
2
2

r x
=
3 r = 13

tan 1 3 =

2
O arct(2)
=

Parte
particular (X+) /+
A +B
le+Se
+

=
=

Art 2) B + +
-

Y(A) +1)
+

b)
+

13/A+ 3)
+
+
f
=

calf: A -
4A 131
+ v
=
10A =
1 A =
4B 133 3
2( 103 0B
5
+

3
calfind=2A
=

4A
=
- - +
=

- +

Xt Al es (f)) An (esunact (+))

con fact
se
=
+
+
Gemplo
&

- Parte
particular (Yn) A
yn
yn
=

A
18/A 0
-
=
-

yn
8yn -
P
1A
-

- =
-

80

A 004
Parte
An18
=

homogenlar -Inn =

m -

11 0
=

yn An8)
=
80p
+

m =

S Falta
hallar constanter

15000 A 800 15000 An 16000 -

80
yo
=
= =
+

Yo 3
=

An (8) 800 0
=
(1500D -

800/18/980p +

0
=

Lubtema:Stabilidad en
ecuaciones difsencia (r 1

la es
ecuación estable
i line
y -
vt=0
{ vi r 1
=

raicer la citabilidad
depende nota de
cuando
hay complyan a

Teorema: mi am+ b
+
0
=

la ecuación
en estable
i

in a 1+b
ue deben ampli abauf
jib 1
 Gemplar 1:X + 1 +
+ 2X
-
=
y
S Xo 1
=

Pate
homogenea: /en Anf =

particular: (X -(
Parte A
=

X
+
+
1 8x
+

=
+0
A +

2A
=
5

m 1
+

0
=

A e
d e

m
=

-
3
↑

Pate

Solución Xf=An +
Yo 1
=
-
Arf -

2 20 +
1
= -An
2
+

=
1An =

-
Gemplo 2: Xf 2
+
-

4xf 1
+ 4xy
+
2
=

Porte arte 2/z-)

homogenia particular (X-) 0 +

m2 y 0
2/25y y(0f 1225+ y) y(+ (y 37
=

ym
=

crf
+

+
+
- - +

(m 2((m 16
-

- -

2) 0
=

carf 2 - 8 y
+
1
=

1
2 multiplicidad
dow 12C =

m =

An(2) Azt(z)
c
z
=

(x + (n =
+

Lalucian:X-An127 Az+
+
r +
ra
1
Xo 1
A,
=
=

12() 2Az
1 02An 1 8A =
-

xn
=

+
= =
+

Ejemplo 3:

+ 10
ary Sy 6y
+

=
-

+
+

1
=
+
+

+z

5
by ++z
-

4y +
=1 8yy
+
=

orf Parte
hamogenea Parte
particular (+(p A
=
+
+B Cajo con caf indepen
m2 - SM +
6 0
=

A(f 2) +
B
+ -

s/A) 1) +
B)
+

6/A/) B)
+

+
+ 10
=
+

(m -

2((m -

3) 0
=
A -
5A 61
+
1
=
- 21 1 1 =
1
=

SA 53 63 1 -
3A 2B 10 =

2A
-

m B
+

3
+

m 2
-
= -
=
+

(x+)n An (24) 123

=
+
-b -

2 2B 10
+
=
3
2
=

Calucian:X An(24) An 134 =

+

1+ +
+
 5
by ++z
-

4y +
=1
+

8yy =

facil
Pate
homogene
Parte
particular /x) A
= cada no
hay
m2 -
4m 8
+ 0
=

A - YA 8A S
+
=

m
+
= = x
518) 5A S =

A 1 =

m y = 1 - 16

m 2
=
2:

r
22
=
- 22 r 25
=

0
fan G 1
4
=
=

Xn=ante cant) azlsam)) +

Lalucian X + An1(an(+)( =
=

A2(28-(+))
+
+
1

Gemgla:

un 0 deba toda la deudar

yo 300D años
=

000
( 0.15/yn
-

yn
+

· =

1
+

Grmplas:
4+xf x+
+1x + 4X++1 4x 12

E
2 =
y byy
-
- + +

X+ 4 x
= +

1
=

af
= + -
+

Pate
y +1 20 xf+ 3yf homogenea
+
- +
=

1 x++1
m2 -
4m y
+

0
=

2 4 y 1
-
+ +

+
=
+

2x+ +
2 4 = +
xf c
+
-
f20 x +
+ +

by+) (m -

2((m -

2) 0
=

X +
+
2 4
=
x+
+
1
+ 20
+
-
x - +

3yf
m 2
=

mutinlicidad 2

3 Voz 4
=
x 2
+

x -
34 x
-

xxx) Anda+Antat =

X +2+
4
= x++
+

1
+
20 - x
+
- 12 -
3x +3x++1

X 2 12 4x + 1 4xf
=

+
+ + -
+

X +
+

2
-
4X +1
+
4x
+
=
+
12
 Parte (X)
particular =
Lalucian
A X+ A,
=

(24 A2+(24) + 12
+

xf An(2+ 1) Az(f +1)(2+ 1)

+

4x 12
+

X +
+

2
-
4X +1
+
+
=
+
+
= +
12
+

A -
YA 4A
+
12
=

A
= 12

Lalucian yy 4 +x
= -

x7 1
+

An(2+ 1)
+
An(f 1)(2+ 1)
+

An(24 1) (24) 12
12
- - +
-

4
+

yy
= + +
+

*
orna Matricial

E
X+ 1 4 X yf
af -
+
= +
-

y +
+
1 -
= 20 xf
+

+ 3yf

-fie
1 --
x +
+

1
-

y+ 1
+
1 3

12 4x
+ 1)
+

0
=

14 2)(X 2) 0
+
+ =

D =
-
2 multiplicidad 2

↑ -- e) ( e
2 1
-
-x - y
=

=
=

1 S Xf
y=
-
=

3 xy (
=
y)(
-

-
2) x (i) (2)
=
+

p(
+
-
2
X2* k
=
+(n+ e(rt
+

/A -
rIP K
= (rehallar

P2) 72
t)(i) -
-
P -

Pn
=
- 1 (1 -
-
1

-
- -
=

P1 P2 +
1
=

- 1 02
+
-

P2 -
=
1

d P2 0Pr 1

ti)ti
=
=

Pr 1
= -
02

x* 2 1 4)f(2 (8)( z
-

P AnfaltartAn/filesartfrer
·I
s
urificar
oriente
-i- toma espinan
x +
+

1
-

y+ 1
+

xt (i)) 2
=
-

fitarte-no filmartfief
(2(( zt (22)(- at (i)
*

-
=

x* 2 ( (+ f 2
=
+

(6)fz
*
Misterfert(6) fart_(.tefart (8fat -

Lolnian
3(5) 12
6 llamo
particular X, Xtor.... A
=

X+ 1 +
4
=
X
+
-
-
yf 3 =,y -
-. . . . =

y +
+
1 -
= 20 xf
+

+ 3yf

0 4
-

BB 4 =

E
A 4 A 3
=

-
= +

A 33 A 12 4 -
8 1 A 12
-
=

3 20 20
+

4
+ - =
+
-
= + + -
=

Lolución general

/:anfra az/fffartff)
7

Dragrama de Fal

E
4 x- y=
afX + 1 = + -
+

y +
+
1 -
= 20 xf
+

+ 3yf

· AXt 4
= +
xx -

yy - x+ -
1X+ 4
= -

y+
-
(X+ 0
=
-
0 4
=
-

y -
=

y=4 =

·
sy + = -
20 xx
+ +

3y +
-

yy
-
byy -
= 20 xf 2yy
+
+
-

1yy 0
= 0
-
=
-
20 xy
+

zyf
+

y1
=

y= 10
2x=
-

12 -
14f

--

1x p(12,4)
19
·y 10
= - =

8
illa
-

punto
6
1xf
- -
=

6 -
*
X+ 12 -
bAx
=

r
-
2-
4
f

↳
ese in i2 x-
 Eercicio 6
· Parte
particular

b)(x 2x
-1 8 + -
y=
(x+) 108
=
7

A
=

y 13 x
1y= 1x =
27
=

+1 -
- X +2
-

xx
+ +
+

+ 1
-
=

1x +z 0
zx +1 y
+ -

+
+ +
+

A A
1A
=

-
+
27
=

IA
27
2X
=

15
2x1 1
8 x +
+2 +
1yt
= -
+ +
=
+

A 108
=

yy 8 +
2x x+
= -

+ 1

xf 23 +
2x 1x 118 ex x 1)(x) A(e Ax(z +108
+

+ =
+ -
+
= +
+

xf z +
237
=

2x++ 1
-
xy 4
+

2x1
+
-

2X +
Xt x =
ane a +
108
+

x +1 1x1
X+ 27
-
+ +

2
+
=

x +2 x= 1
+
x - 27
+ 1
- =
+

Pate
homogena (xn An(e Az(2)
+

y =8 2x x +
.
+
+ -

8 +
1
m2
1m 3

-y z/A/z) Ante+ 108) (ant

- +

0 + -
+

=
(m
=

2((m 2) -
0
=

Az(+ y+ 1) 108)
+

+ +

multiplicidad
m
1
=
2

Drag rama de I al no eta

a escala

b)(x5 2x
x1 8
=
+ -
y=
yd
=

y 13 x
1y=
=

+1 -
-
+ + -

5 e

80 -

AX =0 x=
2xx yf
+ - -

78 -

1X + 8 AX + 0
1x - yy
=
= +
+
B
60 -

0 =
8
1x
+ -

y = SD -

=8
zx
40
y
=
+

30 -

AX
Ay + -
= 15 x+
+
-

y 1y
+
=
+0

20 -

15 x-
3y=
0 -
=
+
-

10
=
SY
10
yy
2x=
=
+
=

⑰ ↑. N & P
ino
↑ N N
18 23 30 yo So so is o o no xf

8
2xt 10
2x 0 1(108) A
+
*
+ -

y
+
=

=
+

metable
solución
18
1xf 62
-
=

y =
=

-
**
x + 108
*
=

/
p(108,62 ↳ T
-
* -

->
*
 Forma Matricial

E
X + 1 +
8
= + 1.5X +
-
yf
b
=+1
y
-
=
1) xf
+
-

0.5y=

↑ =2 (je)fis
-
1
*
-

15 y
4
①(A x I) (re eta- e)
-

1
=
+
-
-

x2
24 1 1 0
(8 41 5 4) 2
=

1
- +
+
- - + - +

- =

↑2 - 4
1
+
= 0

(x -

2114 -

2) 0
=

multiplicidad
X 21 =
dar

o /(i) ex--3 e-
=ye 3f
=

x
+ 2yf
=

x* )(
= x I =
14 -( ere +

↑hatter x2 (Formula:(A -rI/P k) =

P 12
(e)(
22 - =

1
Pr 292
+

P 12 2(1 2011 22 13
a)(o)
22 - -se
-
- -
= · =
=

1 4 P 1/2
Pr 292 2 -
-
+

2 To
=

52 2 2
-

(1
=
-

· = 1 -

22

31 1
3 P
E
-
- =
=

S
hamogened
volución

(de An)(*)de los

xx
est
=An
+

y
n
=

6
particular
volucion

-
A
2A
B 3 15
1
+

A 0
-
-
= + - =

15 A
A 8 B B 1B -
= +
+

-
-
=

15 A
23
-
= +

A =
-
16 2B
+

3 10
2A
+
-

A 16 20
A
+
= -

B 62
-

A 108
IA
-
-
=
36 =

7 valucion

Se An
=

re-Andes fino retr

 levar
Ericio

E X1 1 2x 2y=
x
=
+
-
3

x 1
y +1
=
=
=
+

yy -

=(i):fe
(A - x= 1

irrer-se=ex 3
=
valer prapiar
↳*e x
2
-
34 2
+
-
2

2
x -
34 0
=

1) n0 =

↑ 2) 2xx
aye e

(5) (28) ()
=

3 -

x =-+

y=

2 x3 =

zy=e

-2/(exe 1= (i)
+

3
1X+ 2yf =

x f 2y=
=

Planter valucian
hamogenea

b) anfiflor An
-

frort e

Planter valurian
particular

se s 2 -ti
A 2A =
+ 23 -
3

3 A 2B
= + -

A 3
=
-

2B B 2 =
-
B

A 1 =
3 1
=

Solucion

by an
=

fi) dorftan fort- (I

 Taller S

Gempla 1/si nuede hacer nor matir?

3.4 Porte particular
E
x +1 +
ay
=

+
=

as4 347
1
+

5.4 abxy
+

bx + X= 2
-
=

y
+

+1
+ = +

X +z abXy 4f(as 12)

+

16 ab
=
-
+

caw

(x + ) c(++
1
3.47
+

1x
ay++
=

+2
+

+ =

- 1
cretty) -

ab((44) 4-(as 12)

= +

s.4%)
+

2xf z a(bX + 34
+

abr(4t)) 4-(as
+ = +

c(yf.16) -
=
+
12)

abc(4t) 4f(as
1
X + 34) 16/44)
+

+2
+

abX
=

+as
+
-
=

12)
+

as4 347
1
+

abxy 16 - abc as 12
+

X= 2
+
-
=
· =
+

Parte c(16 -
ab) as 12 +

homogenea
=

m -
ab 0
=
2 as
=
12
+

m ab
= 16 - ab

(Xn Arab =
(x + ( as
=
+

124
ab
Arrab)
16 -

General:X as
12
+
+
=

16 -
ab

Arrab
+ 1

xf 1
+
=
as 12(+++
+

16 -

ab
Solución y
x +1
+
ay
=

+
=
3.4

abtieaster
x 3.4 en e 3. 4
y=
=

1
=
+
f
a a

a
1

y A1rab)*b 12(4) ) 3.4

+

=
+
as +

a(16 -

ab) a

Vrificar
eabilidad a estable
is rab1-> line 0
=

y 8
-

Sumpl2: X + z
+
(012
+
-

2/kf +
+
(1 -

0)k =
= d
condiciones
a, B, o >

m2 (0/a 2/m
+
-

(1
+
-

0) 0
=

m
-
=

(0/a 2) -

(op/a 2 -
-
4 -

5)

2
 rop/a- 2 -
4/n- vp) 0 condicion para que tenga vaia
camplear
ropa -
zy(n -

N =
1-0(a 2) -

+
(rora a) - -
4 -

g)
11ora 2) (rora z) -
+ - -
4 -

gp)) 1 <1

para
condicion en estable

Semplo 3

↑X mAn/0.64 car/artro.8 f) Az (0.64 martes

1xe 2y
E
+
=

x++ 5
=
=
- + +

y +1+ 10 -
x +
yf -

Parte
=

particular
x 3
cam =10 Yo: x +2
8x 2x +
4
-

+1
+
+ =
+

1X + 5
1X y (+) A
+

2
- +

+1 1
+ =
=

+
+ +

2 X
+2
-
=
5
1X+
+

+1
2/10
+ -

xy yf)
+
A -

GA 3A
+ 4
=

X 5
1x 4
3x+ zy= 2A 4
+

-
+2
+

1
-

+ +
+

= + =

3
1x- A 10
5y x
-
+
=
f =
1
+

Lalucian
y= 2x+ 3 5x Az/0.64hem artesas
An) 0.6 ycar/art
+1
+
-
y

X = ro.o f +
=
3

Xf z
+ =
-
5
1x
+

1
+ 4
+
-

zx +(2x+
-

+1+
3
-

gx)
X1 x 15 =

fX
+
+ y
+
-

zX x
+
+

+1
-

8x=
xf
fx 2x- con arte. + Az (0.64en artes tanno
X Antro.
4+
=

z
-
+
+
+1

x +2
8x 2x
4
-

+1 =
+
+
+

Parte
homogenea: Lalucian y. yy 2x++1+3- 8f
=

m2
5m 150 I
f 1

car/artro.a + Az (0.64en artes tan no

E/An) 0.64)
+
-
+

+ es
y=
=

m
= 2

m =

=500
2 -

S/An 0.6 cas/artro.so) Az (0.sthem jarles +

as

2.0: Encontrar
Valate COM x 10
=

Yo:
3
5
=

m =

An0.64car/artro.so) Az/0.64
m 0.505;
artesof o
= =

10 + am
t

r 25/64 0.255081
An(1cow(or) Az (1 umor) +
+

10
=

=
+

r 0.6487
+An0
10 An 10
=

+
=

fanG 0.50 0.81440

An
=
=
=

I As 0
=

O art (0.8144
=
 I
f 1

Az (0.64hen artes tan +

E/An)
+

y=
=

0.64) car/artro.gif +
2
-

S/An 0.6 cas/artro.so) Az (0.sthem jarles +

as

I
f 1

Az (0.64hen artes tanto

E/0 (0.64)
+

car/artro.gf
2
=
+ +
y=
-

S/0/10.6,casarct ro.so) +

Az (0.Sihem jarlesas

s
/A./ro.si nrartos aoff s
=
-

(A/ro.si autoser=10) un

s
//A)
=

(0.st. unrantro.
- (Ackamor off +

3
2A20.64aenrart(0.81) 25
2 -
= + +

0.64umart (0.81) 125

S
2Az
+
=

fart ro.
-18 Az =
0.64um

ro.
A20.64nen fart
-

-
525 Ac
=
·busser carficiente
indeterminador
32

un farst (0.81

I -

Matricial X e.s:
orna
G =
=

2y
E
X+ n
+
=
-
5 +
1X +

y +1+ 10
=
-
x +
-

yf -e.s: es:(
1

-we as ↑ y no
-
S
X + 1 +

y +1
+

(y 4 (e x e)) +
-

-(A xe=
- -
-
=

1
IX x 4
z
-
-
0
+
+
=

4
-
x
4 13 0
+
=

2D

x
5
=

-
=

x
=
=
- sorso
2

4
= 0.505:
=
=
Gemal +2f+im raff
Parar X + 3
+
-
5X+ z
+
8x
+

+ 1
+
- 4X =3.2
+

El caracteristico er:
palinamio
m3 - Sm2 + 8m -
4 (m
= -
212(m -
1)

ar ecuación
escriba parte hamogenea
Xen= andarf=autraf+ab fat

b) ecuación
ucriba parte
particular
3.2 +2f imraff
5X 8x 4X+
+

x +3
+
-
+ 2
+
+

+1
+
-
=

(x (p
+
A
=
2(28
+

B +c
+
+

Dinrat)
+
f
+ cauraf
 Taller 3

Gemplo e

1x -
E
2
X Solución
=

+1
=
=

1
1y
=an +anfe)-fre
x - -
y
+

+1
=

=
+

-- -
~

Dragrama de fall
1x -
E
X 2

y
=

+1
araxeri
=
=

teneren arte
1
y +
+1
x -
=
=

1y -
+

x
-

1
-

2 1x + + 0
=

1X 1x 2 xy -

1X+ 2
-

+
= - = -

12 -
1 3
=

rx ((x
- -

z) 3
=
1X+ 0
=
-

0 -
=

1x+ - 2 -

1xx =
-
2

· X+ -
4
X
=

3
1
=

1 = 1
0 - 0 - -

by + x
zy - 1
y= x+ 3y
1
-
- -
-

y +

( i)(8)
= =

x +
=

3=e
-

Ayy 0 x+
zyy 1
zy x 1
- - -

x
- -

+
-

0
= =

= y=
=
=

yy 2x 3
15 (3)
=

-
-

(2)
·

3
= =

vn
-

y 2( 4)
z 6 2 1
- -

=
-
-
- = =
+

+X
2
=
-

↑ ie)(i)
x
=
=
x
e p( 4,
-
-

18/3)

AXf yf

/-(y = // v
>

-
3-
P/0,0
1
i 3 2
- -
-

& i A ③

Parte Xf
homogenea
>

*
·

sy
/n an anfi =
+
..* y ->

it
Parte particular ->

IA
=-
A = -
2

B A
13 1
=
- -

S
IA 3
1B
-

-
A 2 =
+ =

IA 2
=

IB =
-
S
** gr -

1
A -
= 4 3 =

-
# ->

=fi
 Semylo 2 no hay particular

Es
1

↑= s
ia-
ce= f= roeriers

12 -

6x -
2x x2
+
3
+

0
=

x -
84 17
+
0
=

x 0 =
7768
=

2x
x 0
wi
=
-
=
=

x 41j
=

r 1
1
=

N =
17

tan 0
1
=

O arct(f
=

3 x 4
=

i
+

↑ ---...(je)- :y=
1
aix- 0
-

3 =

1 =
=

15) (y (re =
3-
=

* X 4
=
-

⑮5 - - (8:)
-
-
=
 Semplo 3

3x - y
E
X +1+
=
+ =

+3

0xf 3y +
-

2
y+
-

n
+
=

1
X

8 8: fu
+1
+

y +

+1

& 3
-
4 - 1

(A -

411 =
(13
= -
x)) -

3 -

x)) 0
+

0 -
3 -
A

- -
3x + 34 + 42 0
+ 0
=

2
x 0
=

x 0
=

Cava x 1
=

↑
3- 18)(

3- y e
(5) (5) =
3
=

()
-

S5/n Anfor =

Parte particular
3x - y
E
X +1+
=
+ =

+3

0xf 3y +
-

2
y+
-

n
+
=
 Ereicio de dar
forma matricial

lefter for
(Xx x 3x1 18y=
↳-Anfiter-
An
=
-

y +
+
n 2x7
=
-

0yf
S

↑ ↑ :
3
X =
+1 =

2
y +
+1

!* *
3
/A x I ((3 x) 0
x)) 36
- -

- - =
- +

1 -
27 -
3x + 0x x2)
+
36
+

x2 6x
+
-

27 36
+
0
=

x2 6x
+ + 0 0
=

( 3)(x 3) + +

X =
-
3 mutiplicidad 2

3
x = -
3

) 2x + 6y =3
-
- =

X =
3yf
y

b) 1) 3 = =

=(2) ry
x (2/13 x (3 p+
+
+

↑Unificar (je== (5)

/fort =2 (?) fort

filterter etat =

(i)(-ert (=)(- yf
=

S Encantar x * (A -
NI/P K
=

=)(8) 2 =

6(1 -
188 3.2/2 322)
=
+
-

62 x
=

291 -
6P2 1
=
1 612
+ -

612 1
=

.
691 3 =
1892
+
12 0 =

Pr 1/2
P1 1
=
3P2
+
=
 Forma Matricial X e.s:
G =
=

2y
E
x+ 1
+
=
-5 1X
+ +

1 es:(
-e.s:
3/8
y +1+
=
10 -
x +
-

yf -

-we as ↑ y no
-
S
X + 1 +

y +
+1

(y 4 (e x e)) +
-

-(A xe=
- -
-
=

1
IX x 4
z
-
-
0
+
+
=

4
-
x
4 13 0
+
=

2D

x
5
=

-
=

4
I
= -

83/80

4
= 0.505:
=
=

lavo D 0.5:
G
=
+

13/8-0.5:
215
0.si/yy
0.5:(i)
xe 1318 -
0
=

-
=

1 30 -

tos - y=
a.
(sesi
y ((38) i)- .)
-
-

We we

t = 25/64 0.255081
+

r =
0.6487

fan G 0.500 = 0.8144

=

I
O art (0.8144
=

=ro.sirt
↑caurant (0.8 f(4) (0.81+f()
x -
new lart
 Cavax
I-0.5:
=

Here-e
215

-
1 1 -
x

-... es:)(i)
↑38+
os:
=

-
x + +
3/8 0.5i/y = +
0
=

x= (318 0.5i/y=
=
+

(5) (rszy =

y
=

-((4 i/y) -

** 10. (un
=
laut (0.81+)(8) +

carranctro.817)

⑤-Anfrosercausant
↑ (o.8 fr()- raut(0.8er(8(()
nn +

An froser un rantros or () carranctro.81+)()

-

=
Porte
particular
(
y
E
xxx =
-
s 1xe
+

y +1 + 10
=
-
x-
yf
+

A s
1A z
-
=
+ +

B 10
= -
A B
+

10 A
0
-

. =

A 10
=

5
f(10) 2
+

-
10
+
· =

10
5 2
= +
-

10
2 23
+
=

2 =
B
1
=

⑤-Anfrosercausant
↑ (o.8 tr(4) -
new rant
(0.8t(8((() +

A
froser un ractrosio () carranctrosit()()-(ii
-