ORDEN SUPERIOR

y’’, y’’’

LINEAL )()(')('')( 012 xfyxayxayxa

HOMOGÉNEAS

0)(')('')( 012 yxayxayxa

NO HOMOGÉNEAS

ph yyyxfyxayxayxa

)()(')('')( 012

COEFICIENTES CONSTANTES

EULER CAUCHY

0''' 00

11

22 yxayxayxa

00''''

2

babyayy

ey x

2

42 baa

iaba2

42

2

s w

2

1

)1('''

m

m

m

xmmymxyxy

2)1()1(4

2)1(

24)1()1(

0)1(

2

2

abam

baam

bammm

i

ryvyuyvyu

''''0''

21

21

Caso 1

ba 42 x

x

ey

ey2

1

2

1

xxh ececy 21

21

Caso 2 ba 42

xeyy 21 xxh xececy

21

Caso 3

ba 42 xwis

xwis

eyey

)(2

)(1

)cos(

)(2

)(1

BsenwxwxAeyececy

h

xwisxwish

Caso 1 ba 4)1( 2

2

1

2

1m

m

xyxy

2121

mmh xcxcy

Caso 2

ba 4)1( 2 21 yy xxcxcy mm ln21

Caso 3 ba 4)1( 2 )(

21

iih

iih

BxAxxyxcxcy

)(xr py ncx

nnxkxkxkk ...2

210 xce xke

xsencxc

2

1 cos

xBsenxA cos

xsenexe

x

x

cos

)cos( xBsenxAe x

ORDINARIAS

dxdxf )(

dtdz

dtdx

dtdy

PARCIALES F(x,y,z)

zyx

,,

IMPLICITAS ),''(' yyfy

EXPLICITAS ),,''(' cyyfy

1er ORDEN y’

VARIABLES SEPARABLES )()(

)()(xgyfygxf

dxdy

)(),()(),(

yfyxgxfyxf

dxxgxfdy

ygyf

)()(

)()(

EXACTAS 0 NdyMdx

dxdN

dydM

Ndy

Mdx cyxf );(

FACTOR DE INTEGRACIÓN

)(xf

)(yfdxdN

dxdM

dydM

dydM

dxdN

dxdN

M

MyNxy '

N

NxMyx '

COEFICIENTES HOMOGENEOS

0 NdyMdx),(),(),(),(

yxNttytxNyxMttytxM

b

a

nba ttt ydvvdydxvyxxduudxdyuxy

;;

LINEALES )()()( 01 xgyxa

dxdyxa )()( xfyxP

dxdy

dxxP

ex)(

)( linealecx .)(

Exacta

BERNOULLI

dxdyyndw

ywn

n

)1(

1

sepnlinealn

.var10

nyxgyxadxdyxa )()()( 01 nyxfxP

dxdy )()(

)()1)(( xfwnxPdxdw

dxnxPex )1)(()( linealecx .)( Exacta

COEFICIENTES LINEALES

0)'''()( dycybxadxcbyax

p. paralelas

p. no paralelas

),(',' baba ybxazbyaxz

byaxz

''

),(',' baba kYyhXx

NO LINEAL CAMBIO DE VARIABLE

dydzzy

dxdzy

dxdyz

'

' '2'' yxy

COEFICIENTES INDETERMINADOS

VARIACIÓN DE PARAMETROS 21 )()( yxvyxuy p

Wronskiano

ryy

v

yy

ru

yyv

yyu

0'

'

'0

'

'

'

'

'

1

1

2

2

2

2

1

1

dxvydxuyyp 21