Segun Unidad.docx

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3.

3 y ' v +8 y '''+6 y ''=( x 3 −6 x 2 +12 x−24 ) e−x 3 m4 +8 m3 +6 m2 =0 2 2 m ( 3 m +8 m+6 ) =0 −8± √−8 4 √2 m1,2=0 ; m3 = ; m4 =− ± i 6 3 3 4 − x 3

Y h =C 1 +C 2 x +e

[

C3 cos

(3√2 x)+C sen (√3 2 x )] 4

Para Y p −x

Y p =( Ax 3 +Bx 2 +Cx+D ) e 2

3

−x

2

−x

Y p '=( 3 Ax +2 Bx+C ) e −( Ax +Bx +Cx+D ) e

Y p ''=( 6 Ax+2 B ) e−x −2 ( 3 Ax 2 +2 Bx+C ) e−x + ( Ax 3 +Bx 2 +Cx+D ) e−x 2

3

2

Y p '''=6 Axe −3 ( 6 Ax+2 B ) e +3 ( 3 Ax +2 Bx +C ) e −( Ax +Bx +Cx +D ) e −x

−x

−x

−x

Y p 'v =−24 Axe−x −6 ( 6 Ax +2 B ) e−x +4 ( 3 Ax 2 +2 Bx+C ) e−x + ( Ax 3 +Bx 2 +Cx+D ) e−x v

3

2

3 y ' +8 y '''+6 y ''=( x −6 x +12 x−24 ) e −x −x 2 −x 3 2 −x ⇒−72 Axe −18 ( 6 Ax+2 B ) e +12 ( 3 Ax +2 Bx+C ) e +3 ( Ax +Bx +Cx+D ) e −x −x 2 −x 3 2 −x +48 Axe −24 ( 6 Ax+2 B ) e +24 ( 3 Ax +2 Bx +C ) e −8 ( Ax +Bx +Cx+ D ) e −x 2 −x 3 2 −x 3 2 −x +6 ( 6 Ax+2 B ) e −12 ( 3 Ax +2 Bx+C ) e +6 ( Ax +Bx +Cx+D ) e =( x −6 x +12 x−24 ) e [ Ax 3 +Bx 2 +Cx +( D−24 A ) ] e−x =( x3 −6 x2 +12 x−24 ) e−x A=1 D−24 A=−24 B=−6 D=0 C=12 3

2

−x

Y p =( x −6 x −12 x ) e

−x

4 − x 3

∴Y =C 1 +C2 x+e

[ ( ) ( )] C3 cos

√ 2 x +C sen √2 x +( x 3 −6 x 2 −12 x ) e−x 4 3

3

4.

y '''+2 y ''=( 4 x 2 +6 x−1 ) e 2 x m3 +2m=0

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