Lista 8 de Edo Resolvida

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    28-Apr-2015

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Lista 8 de EDO - Equaes de Ricatti

Resolva a equao de Ricatti dada; y1 uma soluo conhecida para a equao. a) dy = 2 y + y 2 , y1 = 2 dx Q(x) = -1 R(x) = 1 y1 = 2

P(X) = -2

( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x )+ 2 R ( x ) y1 ) dx ( Q ( x )+2 R ( x ) y1 ) dx R ( x)dx w = c1e e e ( 1+ 2 (1)( 2 ) ) dx ( 1+ 2 (1)( 2 ) ) dx ( 1+ 2 (1)( 2 ) ) dx (1)dx w = c1e e e ( 1+ 4 ) dx ( 1+ 4 ) dx ( 1+ 4 ) dx dx w = c1e e e

w = c1e 3 x e 3 x e 3 x dx w = c1e 3 x e 3 x ( w = c1e 3 x e3x ) 3

1 3 A soluo geral : 1 y = y1 + w 1 y = 2+ 1 c1e 3 x 3

b)

dy 4 1 2 = 2 y + y 2 , y1 = dx x x x 1 x R ( x) = 1 y1 = 2 x

Q( x) =

( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x ) + 2 R ( x ) y1 ) dx R ( x )dx w = c1e e e

w = c1e

1 2 + 2 dx x x

e

1 2 + 2 dx x x

e

x +2 x dx

1

2

(1)dx

w = c1e w = c1e w = c1e w = c1e w = c1

1 4 + dx x x

e

1 4 + dx x x

e5

x + x dx

1 4

(1)dx

5 dx x

e

5 dx x

e

x dx dx dx

dx

5 ln x ln x5

e e

5 ln x ln x 5

e e

5 ln x

ln x5

1 1 5 x 5 dx 5 x x 1 1 x6 w = c1 5 5 x x 6 1 1 w = c1 5 x x 6 6 6c x w= 1 5 6x A soluo geral : 1 y = y1 + w 2 1 y= + x 6c1 x 6 6x5 2 6x5 y= + x 6c1 x 6

c)

dy = 9 + 6 y + y 2 , y1 = 3 dx R( x) = 1 y1 = 3

Q( x) = 6

( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x ) + 2 R ( x ) y1 ) dx R ( x )dx w = c1e e e ( 6+ 2 (1)( 3) ) dx ( 6+ 2 (1)( 3) ) dx ( 6+ 2(1)( 3) ) dx (1)dx w = c1e e e ( 66 ) dx ( 66 ) dx ( 66 ) dx dx w = c1e e e

w = c1e 0 e 0 e 0 dx w = c1 1 1dx w = c1 x

A soluo geral : 1 y = y1 + w 1 y = 3 + c1 x

d)

dy = 1 x y + xy 2 , y1 = 1 dx R( x) = x y1 = 1

Q( x) = 1

( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x ) + 2 R ( x ) y1 ) dx ( Q ( x ) + 2 R ( x ) y1 ) dx R ( x )dx w = c1e e e ( ( 1) + 2 ( x )(1) ) dx ( ( 1) + 2 ( x )(1) ) dx ( ( 1)+ 2 ( x )(1) ) dx ( x)dx w = c1e e e ( 1+ 2 x ) dx ( 1+ 2 x ) dx ( 1+ 2 x ) dx dx w = c1e e xe

w = c1e ( x+ x ) e ( x+ x w = c1e x x e x x2 2

2

2

)

xe

( x + x2 )

dx

xe

( x + x2 )

dx

A soluo geral y = y1 + y = 1+ 1 w 1 c1ex x2

e

x x 2

xe

( x+ x2 )

dx