Problem 23 - Advanced Engineering Mathematics

Question

Fill in the blanks or answer true/false. $$ \text { If } \mathscr{L}\\{f(t)\\}=F(s) \text { and } k>0 \text {, then } \mathscr{L}\left\\{e^{a t} f(t-k) \bullet(t-k)\right\\}= $$____

Step-by-Step Solution

Answer

\( e^{-ks} F(s - a) \)

Step 1: Understand the Laplace Transform Shift Theorem

The shifting theorems help determine the Laplace transform of products involving functions and exponentials. We need the time shifting and the exponential shifting (first translation) theorem here.

Step 2: Apply Time Shifting Theorem

The time shifting theorem states that if \( \mathscr{L}\{ f(t) \} = F(s) \), then for any \( k > 0 \), \( \mathscr{L}\{ f(t - k) \} = e^{-ks} F(s) \).

Step 3: Recall Exponential Shifting Theorem

The exponential shifting theorem states \( \mathscr{L}\{ e^{at} f(t) \} = F(s-a) \). Combining this with the time shift parameter gives a complete formula.

Step 4: Combine the Theorems

Combine the time shifting and exponential shifting results. Therefore, \( \mathscr{L}\{ e^{at} f(t - k) \} = e^{-ks} F(s - a) \).

Step 5: Verify Full Expression

Check whether all transformations apply and adjust the expression for any missing brackets or necessary operations such as multiplications if needed.

Key Concepts

Time Shifting Theorem

The Time Shifting Theorem is an important concept in the realm of Laplace Transforms. It is particularly useful for handling functions that are delayed in time. Let's break it down to understand its application and significance.

The theorem states that if you have a Laplace transform of a function, denoted as \( \mathscr{L}\{f(t)\} = F(s) \), and you are looking at a delayed version of this function, such as \( f(t-k) \) where \( k > 0 \), then the Laplace transform will be transformed to include an exponential factor. In formula terms, it modifies to \( \mathscr{L}\{f(t-k)\} = e^{-ks} F(s) \).

This concept is called 'time shifting' because it shifts the original function along the time axis without altering its overall shape, essentially delaying it by \( k \) units. This shift in time introduces an exponential factor \( e^{-ks} \) in the frequency domain, which accounts for the delay. This theorem is powerful because it allows engineers and scientists to model systems that are not instantaneous, such as delayed signals or activities.

Exponential Shifting Theorem

The Exponential Shifting Theorem is another crucial tool in using Laplace Transforms, particularly when dealing with exponential functions. It provides a straightforward way to handle functions that are multiplied by an exponential term, \( e^{at} \), and it adjusts the Laplace transform accordingly.

This theorem transforms \( \mathscr{L}\{e^{at} f(t)\} \) into \( F(s-a) \), where \( F(s) \) is the Laplace transform of \( f(t) \). The transformation shifts the function in the frequency domain by subtracting \( a \) from \( s \), thus adjusting the transform according to the exponential growth or decay modeled by \( e^{at} \).

Why is it useful?

• It transforms complex problems involving exponential functions into simpler problems.
• It helps in solving differential equations where the exponential term is common.
• It provides insights into how systems respond to exponential growth or decay inputs.

These applications show how valuable the Exponential Shifting Theorem is in practical scenarios, such as control systems, signal processing, and physics.

Transforms of Exponential Functions

When dealing with the Laplace Transforms of Exponential Functions, it's essential to understand how the presence of exponents impacts the transform. The primary focus is on how exponential terms modify the original function and how they are represented in the frequency domain.

Exponential functions, typically expressed as \( e^{at} \), can dramatically affect the behavior of systems. The Laplace transform of these functions is governed by the Exponential Shifting Theorem, which allows these exponential effects to be factored directly into the transform, allowing us to analyze and understand the changes seamlessly.

In the combination with other functions, such as \( f(t) \), exponential terms can highlight growth or decay characteristics within complex systems. By transforming these functions using Laplace Transforms, we garner the following advantages:

• Gain deeper insights into both transient and steady-state behaviors of dynamic systems.
• Simplify complex mathematical problems, making them more manageable to solve.
• Develop a better understanding of how exponential growth or decay rates influence outcome behaviors.

Understanding transforms of exponential functions empowers one to predict and manipulate systems that naturally exhibit exponential behaviors, such as electrical circuits, mechanical systems, and population models.