Transformada de Laplace · Transformada de Laplace transformadas de Laplace usando tablas, ejemplo 3 L{cos(3t)}(s)= s s2+32 = s s2+9 Propiedad del corrimiento de la transformada de

Transformada de Laplace JuanManuelRodríguezPrieto

Transformada de Laplace

L f (t){ }(s) = e− st f (t)dt0

∞

∫


L f (t){ }(s) = e− st f (t)dt0

∞

∫

Ejemplo1:CalcularlatransformadadeLaplacedef(t)=1

L 1{ }(s) = e− st1dt0

∞

∫

L 1{ }(s) = e− st dt0

∞

∫

L 1{ }(s) = limB→∞

e− st dt0

B

∫

L 1{ }(s) = limB→∞

e− st

−s 0

B

L 1{ }(s) = limB→∞

e− s0

s− e

− sB

s⎛⎝⎜

⎞⎠⎟

L 1{ }(s) = 1s


L f (t){ }(s) = e− st f (t)dt0

∞

∫

Ejemplo2:CalcularlatransformadadeLaplacedef(t)=t

L t{ }(s) = e− stt dt0

∞

∫Integraciónporpartes

e− stt dt∫

u = tdu = dtdv = e− st

v = − e− st

s

e− stt dt∫ = − te− st

s+ e− st

sdt∫

e− stt dt∫ = − te− st

s− e

− st

s2

L t{ }(s) = limB→∞

− Be− sB

s− e

− sB

s2+ 1s2

⎛⎝⎜

⎞⎠⎟

L t{ }(s) = 1s2


L f (t){ }(s) = e− st f (t)dt0

∞

∫

Ejemplo3:CalcularlatransformadadeLaplacede

L eat{ }(s) = e− steat dt0

∞

∫

eat

L eat{ }(s) = e(a−s )t dt0

∞

∫

L eat{ }(s) = limB→∞

e(a−s )B

a − s− e

(a−s )0

a − s⎛⎝⎜

⎞⎠⎟

L eat{ }(s) = 1s − a


L f (t){ }(s) = e− st f (t)dt0

∞

∫

Ejemplo4:CalcularlatransformadadeLaplacede

L f '(t){ }(s) = e− st f '(t)dt0

∞

∫

f '(t)

e− st f '(t)dt∫Integralporpartes

dv = f '(t)v = f (t)u = e− st

du = −se− st

e− st f '(t)dt∫ = f (t)e− st + s e− st∫ f (t)dt

Transformada de Laplace Ejemplo4:CalcularlatransformadadeLaplacede

L f '(t){ }(s) = e− st f '(t)dt0

∞

∫

f '(t)

e− st f '(t)dt∫ = f (t)e− st + s e− st∫ f (t)dt

L f '(t){ }(s) = f (t)e− st( )0∞+ s e− st f (t)dt

0

∞

∫

L f '(t){ }(s) = s e− st f (t)dt0

∞

∫ − f (0)

L f '(t){ }(s) = sL f (t){ }(s)− f (0)

LatransformadadeLaplacedeladerivanosvaaservirpararesolverecuacionesdiferenciales

Transformada de Laplace TablasdetransformadasdeLaplace

Transformada de Laplace TablasdetransformadasdeLaplace

Transformada de Laplace transformadasdeLaplaceusandotablas,ejemplo1

L t 3 −10t −1{ }(s) =


L t 3 −10t −1{ }(s) = L t 3{ }(s)− L 10t{ }(s)− L 1{ }(s)

= 3¡s4

−10 1s2

− 1s

LinealidaddelatransformadadeLaplace


L t{ }(s) = 1s2

PropiedaddelcorrimientodelatransformadadeLaplace

L f (t)eat{ }(s) = F(s − a)L f (t){ }(s) = F(s)

L te2t{ }(s)CalculemoslatransformadadeLaplacede

PortablasdelastransformadasdeLaplacesabemosque

Usandolapropiedaddelcorrimiento

L te2t{ }(s) = 1(s − 2)2


L cos(3t){ }(s) = ss2 + 32

= ss2 + 9



L e2t cos(3t){ }(s)CalculemoslatransformadadeLaplacede



L e2t cos(3t){ }(s) = s − 2(s − 2)2 + 9




L e2t sin(3t){ }(s)CalculemoslatransformadadeLaplacede


L sin(3t){ }(s) = 3s2 + 32

= 3s2 + 9



L e2t sin(3t){ }(s)CalculemoslatransformadadeLaplacede



L e2t cos(3t){ }(s) = 3(s − 2)2 + 9

Transformada de Laplace TransformadasinversadeLaplace:Ejemplo1

L−1 2s2 − 3s + 5s3

⎧⎨⎩

⎫⎬⎭= L−1 2s2

s3− 3ss3

+ 5s3

⎧⎨⎩

⎫⎬⎭

= L−1 2s− 3s2

+ 5s3

⎧⎨⎩

⎫⎬⎭

= L−1 2s

⎧⎨⎩

⎫⎬⎭− L−1 3

s2⎧⎨⎩

⎫⎬⎭+ L−1 5

s3⎧⎨⎩

⎫⎬⎭

= 2L−1 1s

⎧⎨⎩

⎫⎬⎭− 3L−1 1

s2⎧⎨⎩

⎫⎬⎭+ 5L−1 1

s3⎧⎨⎩

⎫⎬⎭

= 2 − 3t + 52t 2


L−1 3s −1s2 + 4

⎧⎨⎩

⎫⎬⎭= L−1 3s

s2 + 4− 1s2 + 4

⎧⎨⎩

⎫⎬⎭

= L−1 3ss2 + 4

⎧⎨⎩

⎫⎬⎭− L−1 1

s2 + 4⎧⎨⎩

⎫⎬⎭

= 3L−1 ss2 + 4

⎧⎨⎩

⎫⎬⎭− L−1 1

s2 + 4⎧⎨⎩

⎫⎬⎭

= 3cos(2t)− 12sin(2t)


L−1 5s(s + 4)3

⎧⎨⎩

⎫⎬⎭= L−1 5(s + 4 − 4)

(s + 4)3⎧⎨⎩

⎫⎬⎭

= L−1 5(s + 4)− 20(s + 4)3

⎧⎨⎩

⎫⎬⎭

= e−4 t L−1 5s − 20s3

⎧⎨⎩

⎫⎬⎭

= e−4 t L−1 5s2

⎧⎨⎩

⎫⎬⎭− 20e−4 t L−1 1

s3⎧⎨⎩

⎫⎬⎭

= 5e−4 tt − 20e−4 t 12t 2

L−1 F(s − a){ } = eatL−1 F(s){ }

Hemosusadolasiguientepropiedaddelatransformada


L−1 2s − 5s2 − s + 6

⎧⎨⎩

⎫⎬⎭= L−1 2s − 5

(s + 3)(s − 2)⎧⎨⎩

⎫⎬⎭

= L−1 A(s + 3)

⎧⎨⎩

⎫⎬⎭+ L−1 B

(s − 2)⎧⎨⎩

⎫⎬⎭

= 115L−1 1

(s + 3)⎧⎨⎩

⎫⎬⎭− 15L−1 1

(s − 2)⎧⎨⎩

⎫⎬⎭

= 115e−3t − 1

5e2t

A(s + 3)

+ B(s − 2)

= As − 2A + Bs + 3B(s + 3)(s − 2)

= (A + B)+ (3B − 2A)(s + 3)(s − 2)

A = 115

B = − 15

(A + B) = 2−2A + 3B = −5


L−1 2s +1s2 − 4s +13

⎧⎨⎩

⎫⎬⎭= L−1 2s +1

(s − 2)2 + 9⎧⎨⎩

⎫⎬⎭

= L−1 2(s − 2 + 2)+1(s − 2)2 + 9

⎧⎨⎩

⎫⎬⎭

= L−1 2(s − 2)+ 5(s − 2)2 + 9

⎧⎨⎩

⎫⎬⎭

= e2t L−1 2s + 5s2 + 9

⎧⎨⎩

⎫⎬⎭

= e2t L−1 2ss2 + 9

⎧⎨⎩

⎫⎬⎭+ L−1 5

s2 + 9⎧⎨⎩

⎫⎬⎭

⎛⎝⎜

⎞⎠⎟

= e2t 2cos(3t)+ 53sen(3t)⎛

⎝⎜⎞⎠⎟

s2 − 4s +13= 0

r1 = 2 + 3ir2 = 2 − 3i

Raícescomplejasdeldenominador

s2 − 4s +13= s2 − 4s + 4 + 9= (s − 2)2 + 9

L−1 F(s − a){ } = eatL−1 F(s){ }

Hemosusadolasiguientepropiedaddelatransformada

Transformada de Laplace Ecuacionesdiferenciales

y '+ 5y = 1+ t

L y '{ }(s)+ 5L y{ }(s) = L 1{ }+ L t{ }

L y '{ }(s) = sL y{ }(s)− y(0)

L 1{ } = 1s

L t{ } = 1s2

sL y{ }(s)− y(0)+ 5L y{ }(s) = 1s+ 1s2

(s + 5)L y{ }(s) = 1s+ 1s2

+ y(0)

y(0) = 0(s + 5)L y{ }(s) = 1

s+ 1s2

y(t) = − 425e−5t + 5t + 4

25

L y{ }(s) = s +1s2 (s + 5)

L y{ }(s) = As+ Bs2

+ C(s + 5)

= As2 +Cs2 + 5As + Bs + 5Bs2 (s + 5)

A +C = 05A + B = 15B = 1

A = 425

B = 15

C = − 425


y ''− 2y '− 3y = 1

L y ''{ }(s)− 2L y '{ }(s)− 3L y{ }(s) = L 1{ }

L y ''{ }(s) = s2L y{ }(s)− sy(0)− y '(0)L y '{ }(s) = sL y{ }(s)− y(0)

L 1{ } = 1s

s2L y{ }(s)− sy(0)− y '(0)− 2sL y{ }(s)+ 2y(0)− 3L y{ }(s) = 1s

y '(0) = 0

L y{ }(s) = As+ B(s − 3)

+ C(s +1)

= A(s − 3)(s −1)+ Bs(s +1)+Cs(s − 3)s(s − 3)(s +1)

y(0) = 0

s2L y{ }(s)− 2sL y{ }(s)− 3L y{ }(s) = 1s

(s2 − 2s − 3)L y{ }(s) = 1s

L y{ }(s) = 1s(s2 − 2s − 3)

L y{ }(s) = 1s(s − 3)(s +1)

L y{ }(s) = A(s2 − 2s − 3)+ B(s2 + s)+C(s2 − 3s)s(s − 3)(s +1)

L y{ }(s) = As2 + Bs +Cs2 − 2As + Bs − 3Cs − 3As(s − 3)(s +1)

L y{ }(s) = (A + B +C)s2 + (−2A + B − 3C)s − 3As(s − 3)(s +1)


y ''− 2y '− 3y = 1 y '(0) = 0 y(0) = 0

L y{ }(s) = (A + B +C)s2 + (−2A + B − 3C)s − 3As(s − 3)(s +1)

A + B +C = 0−2A + B − 3C = 0−3A = 1

L y{ }(s) = 1s(s2 − 2s − 3)

B +C = 13

B − 3C = − 23

A = − 13

B +C = 13

B − 3C = − 23

C = 14

B = 112

L y{ }(s) = − 13s

+ 112(s − 3)

+ 14(s +1)

y(t) = − 13+ 112e3t + 1

4e− t

L y ''{ }(s)− 2L y '{ }(s)− 3L y{ }(s) = L 1{ }


y ''− 5y '+ 6y = 1 y '(0) = 0 y(0) = 0


y ''− 2y '+ 6y = 1 y '(0) = 0 y(0) = 0

L y ''{ }(s)− 2L y '{ }(s)+ 6L y{ }(s) = L 1{ }

L y ''{ }(s) = s2L y{ }(s)− sy(0)− y '(0)L y '{ }(s) = sL y{ }(s)− y(0)

L 1{ } = 1s

s2L y{ }(s)− sy(0)− y '(0)− 2sL y{ }(s)+ 2y(0)+ 6L y{ }(s) = 1s

s2L y{ }(s)− 2sL y{ }(s)+ 6L y{ }(s) = 1s

(s2 − 2s + 6)L y{ }(s) = 1s

L y{ }(s) = 1s(s2 − 2s + 6)

L y{ }(s) = As+ Bs +C(s2 − 2s + 6)

= A(s2 − 2s + 6)+ Bs2 +Css(s2 − 2s + 6)

= As2 − 2As + 6A + Bs2 +Css(s2 − 2s + 6)

= (A + B)s2 + (−2A +C)s + 6As(s2 − 2s + 6)

A + B = 0−2A +C = 0

6A = 1

A = 16

B = − 16

C = 13


y ''− 2y '+ 6y = 1 y '(0) = 0 y(0) = 0

L y ''{ }(s)− 2L y '{ }(s)+ 6L y{ }(s) = L 1{ }

L y{ }(s) = 1s(s2 − 2s + 6)

L y{ }(s) = As+ Bs +C(s2 − 2s + 6)

L y{ }(s) = 16s

+− 16s + 13

(s2 − 2s + 6)

L y{ }(s) = 16s

+− 16s + 13

(s2 − 2s +1+ 5)

L y{ }(s) = 16s

+− 16(s −1+1)+ 1

3(s −1)2 + 5)

= 16s

+− 16(s −1)+ 1

3− 16

(s −1)2 + 5)

= 16s

+− 16(s −1)+ 1

6(s −1)2 + 5)

L y{ }(s) = 16s

+− 16(s −1)+ 1

6(s −1)2 + 5)

y(t) = 16+ L−1

− 16(s −1)+ 1

6(s −1)2 + 5)

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪ y(t) = 1

6+ etL−1

− 16s + 16

(s2 + 5)

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪


y ''− 2y '+ 6y = 1 y '(0) = 0 y(0) = 0

L y ''{ }(s)− 2L y '{ }(s)+ 6L y{ }(s) = L 1{ }

y(t) = 16+ L−1

− 16(s −1)+ 1

6(s −1)2 + 5)

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

y(t) = 16+ etL−1

− 16s + 16

(s2 + 5)

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

y(t) = 16+ etL−1

− 16s

(s2 + 5)

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪+ etL−1

16

(s2 + 5)

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

y(t) = 16− 16et cos( 5t)+ 5

30et sin( 5t)


y ''−1y '+ 3y = 1 y '(0) = 0 y(0) = 0

Documents

Transformada de Laplace · Transformada de Laplace transformadas de Laplace usando tablas, ejemplo 3 L{cos(3t)}(s)= s s2+32 = s s2+9 Propiedad del corrimiento de la transformada de