Ejemplo Use transformada de Laplace para resolver la ecuacin diferencial

y''@tD + 4 y'@tD + 6 y@tD = 1 - -t

y@0D = 0y'@0D = 1

Se aplica la transformada de Laplace a toda la ecn.

LaplaceTransform@y''@tD + 4 y'@tD + 6 y@tD, t, sD

6 LaplaceTransform@y@tD, t, sD + s2 LaplaceTransform@y@tD, t, sD +4 Hs LaplaceTransform@y@tD, t, sD - y@0DL - s y@0D - y@0D

Simplify@%D

I6 + 4 s + s2M LaplaceTransform@y@tD, t, sD - H4 + sL y@0D - y@0D

LaplaceTransformA1 - -t, t, sE1

s-

1

1 + s

Se despeja la transformada de Laplace de y[t]

SolveBI6 + 4 s + s2M LaplaceTransform@y@tD, t, sD - H4 + sL y@0D - y@0D ==1

s-

1

1 + s,

LaplaceTransform@y@tD, t, sDF

::LaplaceTransform@y@tD, t, sD 1 + 4 s y@0D + 5 s2 y@0D + s3 y@0D + s y@0D + s2 y@0D

s H1 + sL I6 + 4 s + s2M>>

Se sustituyen las condiciones iniciales y se aplica la inversa de la transformada de Laplace

InverseLaplaceTransformB1 + s + s2

s H1 + sL I6 + 4 s + s2M, s, tF

1

6-2 t I-2 t + 2 t + CosA 2 tE + 3 2 SinA 2 tEM

La Solucin es :

y@t_D =1

6-2 t J-2 t + 2 t + CosB 2 tF + 3 2 SinB 2 tFN

1

6-2 t I-2 t + 2 t + CosA 2 tE + 3 2 SinA 2 tEM

f@t_D =1

6-2 t J-2 t + 2 t + CosB 2 tF + 3 2 SinB 2 tFN

1

6-2 t I-2 t + 2 t + CosA 2 tE + 3 2 SinA 2 tEM

Plot@f@tD, 8t, 0, 5 Pi

Para hacerla al revs,se debe de restar la funcin que multiplica a UnitStep a la funcin.

f@t_D = Sin@tD - HUnitStep@t - PiDL Sin@tDSin@tD - Sin@tD UnitStep@- + tD

Plot@f@tD, 8t, 0, 2.5 Pi

SolveBI16 + s2M LaplaceTransform@x@tD, t, sD - s x@0D - x@0D ==s

16 + s2-- s s

16 + s2,

LaplaceTransform@x@tD, t, sDF

::LaplaceTransform@x@tD, t, sD 1

I16 + s2M2

- s I-s + s s + 16 s s x@0D + s s3 x@0D + 16 s x@0D + s s2 x@0DM>>

- s I-s + s s + 16 s + s s2 M

I16 + s2M2

- s I16 s - s + s s + s s2M

I16 + s2M2

InverseLaplaceTransformB- s I16 s - s + s s + s s2M

I16 + s2M2, s, tF

-1

8H-2 - t + H- + tL HeavisideTheta@- + tDL Sin@4 tD

HeavisideTheta es la misma funcin que UnitStep, solo que en el salto, no est definida

f@t_D = -1

8H-2 - t + H- + tL HeavisideTheta@- + tDL Sin@4 tD

-1

8H-2 - t + H- + tL HeavisideTheta@- + tDL Sin@4 tD

Plot@f@tD, 8t, 0, 4 Pi

y'@tD + y@tD = f@tDy@0D = 0

f@tD = 1 0 t < 1-1 t 1

Se reescribe f@tD en terminos de la funcin escaln unitariaf@t_D = 1 - 2 UnitStep@t - 1D1 - 2 UnitStep@-1 + tD

Se grafic para comprobar si la funcin sigue el comportamiento esperado

Plot@f@tD, 8t, 0, 3

SolveBLaplaceTransform@y@tD, t, sD + s LaplaceTransform@y@tD, t, sD - y@0D ==1

s-2 -s

s,

LaplaceTransform@y@tD, t, sDF

::LaplaceTransform@y@tD, t, sD -s H-2 + s + s s y@0DL

s H1 + sL>>

Se aplican condiciones iniciales y se aplica la Inversa de la transformada de Laplace

y@0D = 0

InverseLaplaceTransformB-s H-2 + sLs H1 + sL

, s, tF

-t I-1 + t - 2 I- + tM HeavisideTheta@-1 + tDM

La funcin y@tD es :

y@t_D = -t I-1 + t - 2 I- + tM HeavisideTheta@-1 + tDM

-t I-1 + t - 2 I- + tM HeavisideTheta@-1 + tDM

La grfica de la solucin es :

Plot@y@tD, 8t, 0, 10

In[1]:= f@t_D = UnitStep@t - PiD - UnitStep@t - 2 PiDOut[1]= -UnitStep@-2 + tD + UnitStep@- + tD

In[2]:= Plot@f@tD, 8t, 0, 4 Pi

In[8]:=-2 s I-1 + s + 2 s s y@0DM

s I1 + s2M

Out[8]=-2 s I-1 + s + 2 s s y@0DM

s I1 + s2M

Escalon Unitario.nb 7

In[11]:= InverseLaplaceTransformB-2 s I-1 + s + 2 s s M

s I1 + s2M, s, tF

Out[11]= H-1 + Cos@tDL HeavisideTheta@-2 + tD + H1 + Cos@tDL HeavisideTheta@- + tD + Sin@tD

In[12]:= y@t_D =H-1 + Cos@tDL HeavisideTheta@-2 + tD + H1 + Cos@tDL HeavisideTheta@- + tD + Sin@tD

Out[12]= H-1 + Cos@tDL HeavisideTheta@-2 + tD + H1 + Cos@tDL HeavisideTheta@- + tD + Sin@tD

In[15]:= Plot@y@tD, 8t, 0, 11 Pi