Laplace Transform Time Shift problem

In summary, the Laplace transform for g(t) = 2*e^{-4t}u(t-1) is e^{-s}*(\frac{2}{s + 4}) * \frac{1}{e^{4}}. The time shift is achieved by multiplying the Laplace transform pair of the function (without the delay) by e^{-as}, where a is the delay. In this case, a=1. However, the exponential in the given function is not time-shifted by 1 to the right, so we can use the fact that exponentials multiplied add their exponents and choose e^b to achieve the needed shift. This results in e^{-s}*(\frac{2}{s
  • #1
NewtonianAlch
453
0

Homework Statement


Determine the Laplace transform:

g(t) = 2*e[itex]^{-4t}[/itex]u(t-1)

The Attempt at a Solution



Essentially we're told for a time shift we multiply the Laplace transform pair of the function (without the delay) by e[itex]^{-as}[/itex]

So here a = 1 (for the delay)

The Laplace transform for e[itex]^{-4t}[/itex] is [itex]\frac{1}{s + 4}[/itex]

Multiplying we should get e[itex]^{-s}[/itex]([itex]\frac{2}{s + 4}[/itex])

However the answer is e[itex]^{-s}[/itex]*([itex]\frac{2}{s + 4}[/itex]) * [itex]\frac{1}{e^{4}}[/itex]
 
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  • #2
Shifting is tricky.

You have used the formula L{f(t-a)U(t-a)} = e-asL{f(t)}.

You need the formula for L{f(t)U(t-a)}.
 
  • #3
NewtonianAlch said:

Homework Statement


Determine the Laplace transform:

g(t) = 2*e[itex]^{-4t}[/itex]u(t-1)

The Attempt at a Solution



Essentially we're told for a time shift we multiply the Laplace transform pair of the function (without the delay) by e[itex]^{-as}[/itex]

So here a = 1 (for the delay)

The Laplace transform for e[itex]^{-4t}[/itex] is [itex]\frac{1}{s + 4}[/itex]

Multiplying we should get e[itex]^{-s}[/itex]([itex]\frac{2}{s + 4}[/itex])

However the answer is e[itex]^{-s}[/itex]*([itex]\frac{2}{s + 4}[/itex]) * [itex]\frac{1}{e^{4}}[/itex]

It's because you must be in the form
[tex]e^{-4(t-a)}u(t-a)[/tex]
Take a look at your exponential. It isn't time-shifted by a to the right. What you can do is use the fact that exponentials multiplied add their exponents and the fact that e^b/e^b = 1, so you can multiply by it without changing your values. So you choose e^b so that you can add its exponents and arrive to the needed shift. The e^b in the denominator is factored outside of the linear inverse Laplace operator and you go from there.
 

1) What is the Laplace Transform Time Shift problem?

The Laplace Transform Time Shift problem is a mathematical concept that involves shifting the independent variable in a function that has been transformed using the Laplace transform. This problem is commonly encountered in the field of engineering, physics, and mathematics.

2) How do you solve the Laplace Transform Time Shift problem?

To solve the Laplace Transform Time Shift problem, you can use the time-shifting property of the Laplace transform. This property states that if the function f(t) has a Laplace transform F(s), then the transformed function f(t−t0) has a Laplace transform e−t0sF(s). This means that to solve the problem, you need to multiply the transformed function by the exponential term e−t0s.

3) What is the significance of the Laplace Transform Time Shift problem?

The Laplace Transform Time Shift problem is significant because it allows us to easily solve differential equations involving time-delayed systems. It also helps in analyzing and understanding the behavior of systems with time delays, which is crucial in many real-world applications.

4) Can the Laplace Transform Time Shift problem be applied to any function?

Yes, the Laplace Transform Time Shift problem can be applied to any function that has a Laplace transform. However, for the time-shifting property to be applicable, the function must be a causal function, meaning it must be zero for negative values of t. If the function is non-causal, the time-shifting property cannot be used.

5) Are there any other properties of the Laplace Transform that are useful in solving the Time Shift problem?

Yes, apart from the time-shifting property, there are other properties of the Laplace Transform that are useful in solving the Time Shift problem. These include the linearity property, the scaling property, and the time differentiation property. These properties can be used in conjunction with the time-shifting property to solve more complex problems involving time-shifted functions.

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