The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True of False?

Name: The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True of False?
Uploaded: 2022-02-23T17:22:57+00:00
Description: The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True of False?

Question

unett · Accepted Answer

Answer: False. The product of the Laplace transforms of the two functions that make up a function is not the function itself.Explanation: Laplace transform of a function f(t) is given by:F(s)=∫0∞e−stf(t)dtThen, for any two functions f(t) and g(t) the Laplace transform of the product of two functions is:L{f(t)⋅g(t)}Integral form is:&amp;nbsp;∫0∞e−st[f(t)⋅g(t)]dtWhich is not equal to&amp;nbsp;∫0∞e−stf(t)dt⋅∫0∞e−stg(t)dtTherefore, the given statement if false.However, it is equivalent to the convolutional product of the Laplace transforms of each function.If F(s) and G(s) are the Laplace transforms of the functions f(t) and g(t), then,Laplace transform of&amp;nbsp;f(t)⋅g(t)&amp;nbsp;is:L{f(t)⋅g(t)}=F(s)⋆G(s)Where, indicates convolution of the two functions, given byf⋆g(t)=∫u=0tf(u)g(t−u)duAs a result, the statement is untrue.

fudzisako · Accepted Answer

The&amp;nbsp;Laplace&amp;nbsp;transform&amp;nbsp;of&amp;nbsp;the&amp;nbsp;product&amp;nbsp;of&amp;nbsp;two&amp;nbsp;functions&amp;nbsp;is&amp;nbsp;the&amp;nbsp;product&amp;nbsp;of&amp;nbsp;the&amp;nbsp;Laplace&amp;nbsp;transforms&amp;nbsp;of&amp;nbsp;each&amp;nbsp;given&amp;nbsp;function.&amp;nbsp;True&amp;nbsp;or&amp;nbsp;False?The statement is false. In general, the Laplace transform of the product of two functions is not equal to the product of their individual Laplace transforms. Mathematically, if F(t) and G(t) are two functions, then the Laplace transform of their product F(t)·G(t) is given by:ℒ{F(t)·G(t)}≠ℒ{F(t)}·ℒ{G(t)}In most cases, the Laplace transform of a product involves convolution rather than simple multiplication. Therefore, the statement is false.

Jazz Frenia · Accepted Answer

Step 1:To determine whether the statement &#039;&#039;The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function&#039;&#039; is true or false, we need to consider the definition and properties of the Laplace transform.The Laplace transform of a function f(t) is defined as:F(s)=ℒ{f(t)}=∫0∞e−stf(t)dtwhere F(s) is the Laplace transform of f(t).Step 2:Now, let&#039;s consider two functions f(t) and g(t), and their Laplace transforms F(s) and G(s), respectively. We want to determine if the Laplace transform of their product, h(t)=f(t)·g(t), is equal to the product of their individual Laplace transforms, H(s)=F(s)·G(s).To do this, we will compute the Laplace transform of h(t) and see if it matches the expression H(s). Using the definition of the Laplace transform, we have:ℒ{h(t)}=ℒ{f(t)·g(t)}=∫0∞e−st(f(t)·g(t))dtStep 3:Now, we can rewrite the integral as a product of two separate integrals:∫0∞e−st(f(t)·g(t))dt=∫0∞e−stf(t)dt·∫0∞e−stg(t)dtThis shows that the Laplace transform of the product of two functions is equal to the product of their individual Laplace transforms.Therefore, the statement &#039;&#039;The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function&#039;&#039; is true.

xleb123 · Accepted Answer

The statement you provided is True. The Laplace transform of the product of two functions is indeed the product of the Laplace transforms of each given function.Let&#039;s denote the Laplace transform of a function f(t) as F(s). Using this notation, the Laplace transform of the product of two functions, f(t) and g(t), can be expressed as:ℒ{f(t)·g(t)}=F(s)·G(s)Here, F(s) represents the Laplace transform of f(t), and G(s) represents the Laplace transform of g(t).To emphasize this relationship, we can write the statement in LaTeX markup as:ℒ{f(t)·g(t)}=F(s)·G(s)Therefore, the Laplace transform of the product of two functions is indeed the product of the Laplace transforms of each given function.

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True of False? Explain why.

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