Laplace
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Publicado: 24 de julio de 2012
TRANSFORMADAS DE LAPLACE
Recopilado y publicado por: Pedro González
Ejercicios resueltos
1. Transformadas de Laplace por definición
2. Transformadas de Laplace utilizando teoremas
3. Transformadas inversas
4. Derivada de transformada
5. Teorema de convolución
6. Ecuaciones diferenciales con condiciones iniciales (transformada)7. Ecuaciones integrales
8. Ecuaciones integrodiferenciales
9. Circuitos
10. Sistemas de ecuaciones diferenciales(método de la transformada)
TRANSFORMADAS DE LAPLACE POR DEFINICIÓN:
1) [pic]
L[pic]
2) [pic]
L[pic]
[pic]
3) [pic]
L[pic]
TRANSFORMADAS DE LAPLACE UTILIZANDO TEOREMAS:
1) [pic]
L[pic]L[pic]+ L[pic]= [pic]
2) [pic]
L[pic] L[pic]+ 6 L[pic]- 3 L [pic]= [pic]
3) [pic]
L[pic]= L[pic]+ 3 L[pic]+ 3 L[pic]+ L[pic]= [pic]
4) [pic]
L[pic]= L[pic]+ 2 L[pic]+ L[pic]= [pic]
5) [pic]
L[pic] L[pic]- 5 L[pic]+ 10 L[pic]- 10 L[pic]+ 5[pic]- L[pic]= [pic]
TRANSFORMADAS DE LAPLACE (1er. TEOREMA DE TRASLACIÓN):
1) [pic]
L[pic] L[pic]2) [pic]
L[pic] L[pic]
TRANSFORMADAS INVERSAS:
1)L-1 [pic] [pic] L-1[pic]= [pic]
2) L-1[pic] [pic] L-1[pic]= [pic]
3) L-1[pic] L-1[pic]+ L-1[pic]= [pic]
4) L-1[pic]= L-1 [pic]= [pic] L-1[pic] [pic] L-1[pic] [pic] L-1 [pic]=
[pic]
5) L-1[pic]= L-1[pic]= L-1[pic]=
L-1[pic]+ [pic] L-1[pic]+ [pic]L-1[pic]+ [pic]L-1 [pic]= [pic]
6) L-1[pic]7) L-1[pic]= L-1[pic] L-1[pic]= [pic]
8) L-1[pic] L-1[pic]= [pic]L-1[pic] [pic]
9) L-1[pic] [pic]L-1[pic] [pic]
10) L-1[pic] [pic] L-1[pic]
11) L-1[pic]= [pic] L-1[pic] L-1[pic]
12) L-1[pic]
[pic]
[pic] , [pic] y [pic]
L-1[pic]= [pic] L-1[pic] L-1[pic] L-1[pic]= [pic]
13) L-1[pic]
[pic]
L-1[pic]= [pic] L-1[pic] L-1[pic]= [pic]14) L-1[pic]
[pic]
[pic] , [pic], [pic] y [pic]
L-1[pic]= [pic] L-1[pic] L-1 [pic]
TRANSFORMADAS INVERSAS (1er. TEOREMA DE TRASLACIÓN):
1)L-1 [pic] [pic] L-1[pic]
2) L-1[pic] L-1[pic] L-1[pic]
L-1[pic] L-1[pic]= [pic]
3) L-1[pic] L-1[pic] L-1[pic][pic]
L-1 [pic] [pic]
4) L-1[pic] L-1[pic] L-1 [pic]
L-1[pic] L-1[pic] L-1[pic] [pic]L-1[pic]
[pic] L-1[pic][pic] L-1[pic] L-1[pic]
[pic]
5) L-1[pic]
[pic]
[pic], [pic] y [pic]
L-1[pic]= L-1[pic] L-1[pic] L-1[pic] L-1[pic] L-1 [pic]=
[pic]
DERIVADA DE TRANSFORMADA:
1)L[pic]= [pic] L[pic]=[pic]
[pic]
2) L[pic] L[pic]
3) L[pic] L[pic]
[pic] [pic]
4) L[pic]L[pic]
[pic][pic]
[pic]
5)L[pic] L [pic]
[pic] [pic]
[pic] [pic]
6) L[pic] L [pic]
[pic] [pic]
[pic] [pic]
[pic]
[pic]
TRANSFORMADAS DE LAPLACE (2do. TEOREMA DE TRASLACIÓN):
1)L[pic] L[pic]
2) L[pic] L[pic]
3) L[pic] L[pic] L[pic] L[pic]
[pic] L[pic] [pic] L[pic] [pic]
4) L[pic] L[pic]
5) L[pic] L[pic] L[pic]
6) L[pic] L[pic] L[pic]
[pic]L[pic] L[pic] [pic] L[pic] L[pic]
7) L[pic] L[pic] L[pic]
L [pic] [pic] L[pic] L[pic]
8) L[pic] L[pic]
TRANSFORMADAS INVERSAS (2do. TEOREMA DE TRASLACIÓN):
1)L-1[pic] L-1[pic] L-1 [pic]
[pic]
2) L-1[pic] L-1[pic]= L-1[pic] L-1[pic]
L-1[pic] [pic]
[pic]
3) L-1[pic]= L-1[pic]
[pic]
L-1[pic]= L-1[pic] L-1[pic]
4) L-1[pic] L-1[pic][pic]
L-1[pic]= L-1[pic]
[pic]
TEOREMA DE CONVOLUCIÓN:
1)L[pic]
[pic]
L[pic] L[pic] L[pic]
2) L-1[pic] L-1[pic] L-1[pic] [pic] [pic] [pic]
3) L-1[pic] L-1[pic] L-1 [pic]
[pic] [pic]
4) L-1[pic] L-1[pic]L-1[pic]= [pic]
[pic]
5) L-1[pic] L-1[pic] L-1 [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]...
Recopilado y publicado por: Pedro González
Ejercicios resueltos
1. Transformadas de Laplace por definición
2. Transformadas de Laplace utilizando teoremas
3. Transformadas inversas
4. Derivada de transformada
5. Teorema de convolución
6. Ecuaciones diferenciales con condiciones iniciales (transformada)7. Ecuaciones integrales
8. Ecuaciones integrodiferenciales
9. Circuitos
10. Sistemas de ecuaciones diferenciales(método de la transformada)
TRANSFORMADAS DE LAPLACE POR DEFINICIÓN:
1) [pic]
L[pic]
2) [pic]
L[pic]
[pic]
3) [pic]
L[pic]
TRANSFORMADAS DE LAPLACE UTILIZANDO TEOREMAS:
1) [pic]
L[pic]L[pic]+ L[pic]= [pic]
2) [pic]
L[pic] L[pic]+ 6 L[pic]- 3 L [pic]= [pic]
3) [pic]
L[pic]= L[pic]+ 3 L[pic]+ 3 L[pic]+ L[pic]= [pic]
4) [pic]
L[pic]= L[pic]+ 2 L[pic]+ L[pic]= [pic]
5) [pic]
L[pic] L[pic]- 5 L[pic]+ 10 L[pic]- 10 L[pic]+ 5[pic]- L[pic]= [pic]
TRANSFORMADAS DE LAPLACE (1er. TEOREMA DE TRASLACIÓN):
1) [pic]
L[pic] L[pic]2) [pic]
L[pic] L[pic]
TRANSFORMADAS INVERSAS:
1)L-1 [pic] [pic] L-1[pic]= [pic]
2) L-1[pic] [pic] L-1[pic]= [pic]
3) L-1[pic] L-1[pic]+ L-1[pic]= [pic]
4) L-1[pic]= L-1 [pic]= [pic] L-1[pic] [pic] L-1[pic] [pic] L-1 [pic]=
[pic]
5) L-1[pic]= L-1[pic]= L-1[pic]=
L-1[pic]+ [pic] L-1[pic]+ [pic]L-1[pic]+ [pic]L-1 [pic]= [pic]
6) L-1[pic]7) L-1[pic]= L-1[pic] L-1[pic]= [pic]
8) L-1[pic] L-1[pic]= [pic]L-1[pic] [pic]
9) L-1[pic] [pic]L-1[pic] [pic]
10) L-1[pic] [pic] L-1[pic]
11) L-1[pic]= [pic] L-1[pic] L-1[pic]
12) L-1[pic]
[pic]
[pic] , [pic] y [pic]
L-1[pic]= [pic] L-1[pic] L-1[pic] L-1[pic]= [pic]
13) L-1[pic]
[pic]
L-1[pic]= [pic] L-1[pic] L-1[pic]= [pic]14) L-1[pic]
[pic]
[pic] , [pic], [pic] y [pic]
L-1[pic]= [pic] L-1[pic] L-1 [pic]
TRANSFORMADAS INVERSAS (1er. TEOREMA DE TRASLACIÓN):
1)L-1 [pic] [pic] L-1[pic]
2) L-1[pic] L-1[pic] L-1[pic]
L-1[pic] L-1[pic]= [pic]
3) L-1[pic] L-1[pic] L-1[pic][pic]
L-1 [pic] [pic]
4) L-1[pic] L-1[pic] L-1 [pic]
L-1[pic] L-1[pic] L-1[pic] [pic]L-1[pic]
[pic] L-1[pic][pic] L-1[pic] L-1[pic]
[pic]
5) L-1[pic]
[pic]
[pic], [pic] y [pic]
L-1[pic]= L-1[pic] L-1[pic] L-1[pic] L-1[pic] L-1 [pic]=
[pic]
DERIVADA DE TRANSFORMADA:
1)L[pic]= [pic] L[pic]=[pic]
[pic]
2) L[pic] L[pic]
3) L[pic] L[pic]
[pic] [pic]
4) L[pic]L[pic]
[pic][pic]
[pic]
5)L[pic] L [pic]
[pic] [pic]
[pic] [pic]
6) L[pic] L [pic]
[pic] [pic]
[pic] [pic]
[pic]
[pic]
TRANSFORMADAS DE LAPLACE (2do. TEOREMA DE TRASLACIÓN):
1)L[pic] L[pic]
2) L[pic] L[pic]
3) L[pic] L[pic] L[pic] L[pic]
[pic] L[pic] [pic] L[pic] [pic]
4) L[pic] L[pic]
5) L[pic] L[pic] L[pic]
6) L[pic] L[pic] L[pic]
[pic]L[pic] L[pic] [pic] L[pic] L[pic]
7) L[pic] L[pic] L[pic]
L [pic] [pic] L[pic] L[pic]
8) L[pic] L[pic]
TRANSFORMADAS INVERSAS (2do. TEOREMA DE TRASLACIÓN):
1)L-1[pic] L-1[pic] L-1 [pic]
[pic]
2) L-1[pic] L-1[pic]= L-1[pic] L-1[pic]
L-1[pic] [pic]
[pic]
3) L-1[pic]= L-1[pic]
[pic]
L-1[pic]= L-1[pic] L-1[pic]
4) L-1[pic] L-1[pic][pic]
L-1[pic]= L-1[pic]
[pic]
TEOREMA DE CONVOLUCIÓN:
1)L[pic]
[pic]
L[pic] L[pic] L[pic]
2) L-1[pic] L-1[pic] L-1[pic] [pic] [pic] [pic]
3) L-1[pic] L-1[pic] L-1 [pic]
[pic] [pic]
4) L-1[pic] L-1[pic]L-1[pic]= [pic]
[pic]
5) L-1[pic] L-1[pic] L-1 [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]...
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