Problem 44 - Advanced Engineering Mathematics

Question

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{\left(1+e^{-2}\right)^{2}}{s+2}\right\\} $$

Step-by-Step Solution

Answer

\( f(t) = (1 + e^{-2})^2 e^{-2t} \)

Step 1: Examine the Expression

We are given the inverse Laplace transform of the expression \( \frac{(1 + e^{-2})^2}{s+2} \). This expression can be written as \( \frac{K}{s+2} \), where \( K = (1 + e^{-2})^2 \).

Step 2: Identify the Inverse Formula

The inverse Laplace transform of \( \frac{K}{s-a} \) is \( K e^{at} \). In this case, \( a = -2 \). So, the inverse Laplace transform of \( \frac{1}{s+2} \) is \( e^{-2t} \).

Step 3: Apply the Inverse Transform

Since our expression is \( \frac{(1 + e^{-2})^2}{s+2} \), we multiply \( (1 + e^{-2})^2 \) by the result from Step 2, which gives us \( (1 + e^{-2})^2 e^{-2t} \).

Step 4: Simplify the Final Expression

Thus, the inverse Laplace transform of the given function is \( f(t) = (1 + e^{-2})^2 e^{-2t} \).

Key Concepts

Inverse Laplace Transform

The inverse Laplace transform is a method used to find a time-domain function, given its equivalent representation in the Laplace domain. It serves as a tool for taking functions that are defined in the complex frequency domain back to the time domain, which is often how we experience or need to work with certain mathematical phenomena in the real world.

When dealing with inverse Laplace transforms, the main goal is to convert a given expression, such as \( \frac{K}{s-a} \), back to the form \( f(t) \). The basic formula we use here is that if a function in the Laplace domain is \( \frac{1}{s-a} \), then its inverse transform in the time domain is \( e^{at} \).

In the problem above, we applied the inverse Laplace transform to \( \frac{(1 + e^{-2})^2}{s+2} \), where \( a = -2 \). This results in multiplying the constant \((1+e^{-2})^2\) by \( e^{-2t} \), thus giving \( f(t) = (1 + e^{-2})^2 e^{-2t} \). This beautifully illustrates how inverse transforms are designed to convert s-domain representations back to their original, time-based form.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the context of inverse Laplace transforms, the base \( e \) (Euler's number, approximately 2.718) is frequently used, particularly when describing growth or decay processes.

An exponential function like \( e^{at} \) commonly arises in physics and engineering problems. Here, \( a \) represents a growth rate if positive, or a decay rate if negative. In our solution, \( e^{-2t} \) signifies an exponentially decaying process, commonly seen in scenarios like cooling or electrical discharging.

Let’s look at some points about exponential functions:

• They are continuous and smooth for all real numbers in their domain.
• Their rate of change is proportionate to their current value.
• In real-world applications, they describe phenomena where quantities increase or decrease at rates proportional to their current size.

Exponential functions are a cornerstone in engineering, modeling numerous real-world scenarios with precision and ease.

Engineering Mathematics

Engineering mathematics is a branch of applied mathematics used to solve problems in engineering fields. It involves various techniques and methods, including differential equations, calculus, and transforms like the Laplace transform, to analyze and design systems.

Laplace transforms, which are integral to this discipline, help simplify complex differential equations into more manageable algebraic forms. This simplification is particularly vital in systems analysis and control engineering, facilitating the understanding of dynamic system behaviors in the s-domain.

Here's how engineering mathematics is applied using Laplace transforms:

• It converts convoluted differential equations into simpler algebraic equations.
• It enables engineers to analyze systems in the frequency domain, gaining insights into system stability and behavior.
• It is widely used in electrical engineering, signal processing, and control theory for system design and analysis.

In essence, engineering mathematics provides the foundational tools engineers need to model and solve problems efficiently, ensuring accurate predictions and designs across various technological applications.