Propiedades de la transformada de laplace

What is the Laplace transform?

The Laplace transform is an invaluable mathematical tool for converting time-dependent functions into functions of a new variable, usually denoted as ( S ). This method is especially useful in solving differential equations, simplifying the analysis of systems in engineering and physics. Through various properties, we can further facilitate the work with the Laplace transform and its application in complex systems.

What are the properties of the Laplace transform?

Throughout this discussion, we will explore four key properties of the Laplace transform that will allow us to tackle problems more efficiently and with less effort.

What is the product by a constant property?

This property is simple but powerful. If we want to find the Laplace transform of a function multiplied by a constant, the constant can be separated from the calculation:

Example: to find the Laplace transform of ( 7 \cdot \sin(8t) ), we identify that ( 7 ) is the constant. According to the property, we only need to obtain the transform of ( \sin(8t) ) and then multiply the result by 7.

The transform of: [ \sin(kt) ] is: [ \frac{k}{s^2 + k^2} ].

Therefore, for ( \sin(8t) ): [ \frac{8}{s^2 + 64} ]

We multiply by 7, obtaining: [ \frac{56}{s^2 + 64} ]

What does the linearity property establish?

The linearity property allows us to decompose the transform of a sum of functions into sums of the individual transforms, thus simplifying the calculation process:

Example: for ( 3t^2 - 9t ), we separate:

1. ( 3 \dot \text{Transform of } t^2 )
2. ( -9 \dot \text{Transform of } t )

Using the transform table:

• For ( t^n ), the transform is: [ \frac{n!}{s^{n+1}} ]

For ( t^2 ): [ \frac{2!}{s^3} = \frac{2}{s^3} ] Multiplied by 3, the result is: [ \frac{6}{s^3}} ]

For ( t ): [ \frac{1!}{s^2} = \frac{1}{s^2} ] Multiplied by 9, gives: [ \frac{9}{s^2} ].

Then, the transform of ( 3t^2 - 9t ): [ \frac{6}{s^3} - \frac{9}{s^2} ].

What does the translation property consist of?

When we include an exponential term, such as ( e^{at} ), in the product with a function, the translation property allows us to adjust the variable ( s ) to ( s-a ).

Example: For ( e^{2t} \cdot \sin(3t) ):

1. The transform of (\sin(3t)) is: [ \frac{3}{s^2 + 9} ]

With the property, we replace ( s ) by ( s-2 ): [ \frac{3}{(s-2)^2 + 9} ]

What does the property of the derivative tell us?

The derivative property applies to functions multiplied by ( t^n ). It allows us to express the transform in terms of the derivatives of the function.

Example: For ( t \cdot \cos(5t) ):

1. The transform of ( \cos(5t) ): [ \frac{s}{s^2 + 25} ]

Derived according to: [ \text{Transform of } t^n \text{ is expressed as } (-1)^n \cdot f^{(n)}(s) ]

For ( t ), derivative: [ -left(\frac{d}{ds}left(\frac{s}{s^2 + 25}right)} ] ]

This operation is summarized using quotient rule: [ \frac{(s^2 + 25) \cdot 1 - 2s \cdot s}{(s^2 + 25)^2} ]

Results in: [ \frac{s^2 + 25 - 2s^2}{(s^2 + 25)^2} = \frac{25 - s^2}{(s^2 + 25)^2} ]

This handling of properties and strategic use of the Laplace transform table are fundamental to solving differential equations quickly and accurately. Don't stop here; practice will make you more adept at their application. As you progress, you will find that solving transforms of this style will become increasingly intuitive. Next time, we will delve deeper into the inverse Laplace transform and its properties. Keep learning and mastering these techniques!