Introducci贸n al Curso

1

Introducci贸n y presentaci贸n del curso

Fundamentos de Ecuaciones Diferenciales

2

驴Para qu茅 nos sirven las ecuaciones diferenciales?

3

驴Que es una ecuaci贸n diferencial?

4

Tipos de ecuaciones diferenciales

5

Conceptos b谩sicos de c谩lculo

Ecuaciones Diferenciales de Primer Orden

6

驴Que es una ecuaci贸n separable?

7

Ejemplo de ecuaci贸n separable

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Procedimiento para saber si una ecuaci贸n es separable

9

M茅todo de sustituci贸n lineal

10

Ejemplo de sustituci贸n lineal

11

Ecuaciones diferenciales exactas

12

Ejemplo de ecuaciones diferenciales exactas

13

Funciones homog茅neas, c贸mo identificarlas

14

Ejemplo de funciones homog茅neas

15

Ecuaciones con coeficientes lineales

16

Ejemplo de ecuaciones con coeficiente lineales

17

Resoluci贸n del desaf铆o

18

驴Que es un factor integrante?

19

Factor integrante caso 1

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Factor Integrante caso 2

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Factor integrante caso 3

22

Ecuaciones diferenciales lineales

23

Ejemplo de ecuaciones diferenciales lineales

24

Ejercicios de ecuaciones diferenciales de primer orden

Ecuaciones Diferenciales de Segundo Orden

25

驴Qu茅 es una soluci贸n linealmente independiente?

26

Ecuaciones lineales homog茅neas

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Ejemplo de ecuaciones lineales homog茅neas

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Ecuaciones lineales homog茅neas con ra铆ces complejas

29

Ejemplo de ecuaciones lineales homog茅neas con ra铆ces complejas

30

Ecuaci贸n diferencial no homog茅nea

31

Coeficientes indeterminados

32

Ejemplo de coeficientes indeterminados

33

Variaci贸n de par谩metros

34

Ejemplo de variaci贸n de par谩metros

35

Ejercicios de ecuaciones diferenciales de segundo orden

Modelos matem谩ticos

36

Creaci贸n de un modelo matem谩tico

37

Crecimiento poblacional

38

Primer ejemplo de crecimiento poblacional

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Segundo ejemplo de crecimiento poblacional

40

Ley de newton de enfriamiento

41

Ejemplo de la ley de newton de enfriamiento

42

Propagaci贸n de un virus y ejemplo

43

Ejercicios de modelos matem谩ticos

Transformada de laplace

44

Conceptos claves para entender la transformada de laplace

45

Introducci贸n a la transformada de laplace

46

Introducci贸n y transformada de una exponencial

47

Propiedades de la transformada de laplace

48

Transformada inversa

49

Ejemplo de transformada inversa

50

Ejercicios de transformada de laplace

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Resources

What is the inverse Laplace transform?

The inverse Laplace transform is a fundamental mathematical concept in system and signal analysis. Its purpose is to reverse the process of a Laplace transform to obtain the original function in the time domain. When working with a function that depends on the variable ( s ) in the Laplace domain, the inverse transform allows to identify the original function ( f(t) ).

How is the Laplace transform table used?

To facilitate the calculation of the inverse Laplace transform, a table is used that relates each function to its corresponding transform. This table, which is indispensable for both direct and inverse transforms, allows one to quickly identify which function in the time domain corresponds to a given transform in the domain of ( s ).

For example:

  • By recognizing that a specific function in the domain of ( s ) is a known result of the table, we can use this knowledge to derive its inverse transform immediately.

What are the essential properties of the inverse Laplace transform?

The properties of the inverse Laplace transform are crucial for understanding and solving problems in engineering and applied mathematics. Two fundamental properties that facilitate this process are:

  1. Property of the product by a constant:

    • If you have a transformed function multiplied by a constant, you can extract this constant. The inverse transform of ( a \cdot F(s) ) is simply ( a \times f(t) ), where ( a ) is the constant.
    L^{-1} \{a\cdot F(s)\} = a \cdot f(t)
  2. Property of linearity or sum of functions:

    • Linearity implies that the inverse transform of the sum (or subtraction) of two transformed functions is equal to the sum (or subtraction) of the inverse transforms of each other. If they are also multiplied by constants, these can also be expressed out.

    [L^{-1} {c_1 \cdot F(s) + c_2 \cdot G(s)} = c_1 \cdot f(t) + c_2 \cdot g(t) ]

These properties allow complex problems to be decomposed into more manageable steps, fostering a deeper and more efficient understanding of the systems under analysis.

How are these properties applied in practice?

To apply these properties to a specific problem, follow these simple steps:

  • Identify: Use the transform table to recognize the parts of the function in the domain ( s ).
  • Apply the properties: Use the property of the product by a constant and the sum of functions to simplify calculations.
  • Solve: Find the corresponding function ( f(t) ) using the tools and concepts described.

These strategies are not only theoretical, they are also applied in fields such as electrical engineering, mechanical engineering and industrial process control. Thus, understanding them is vital for those who wish to develop a robust approach to solving complex mathematical problems. Take heart and continue exploring the fascinating world of Laplace transforms!

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