Problem 11 - Advanced Engineering Mathematics

Question

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s+2)^{3}}\right\\} $$

Step-by-Step Solution

Answer

The inverse Laplace transform is \( \frac{t^2}{2} e^{-2t} \).

Step 1: Recognize the Laplace inverse formula to use

We need to find the inverse Laplace transform of the given function. The form \( \frac{1}{(s+a)^n} \) corresponds to a known Laplace transform result, where \( \mathscr{L}^{-1}\{\frac{1}{(s+a)^n}\} = \frac{t^{n-1}}{(n-1)!} e^{-at} \) for \( n \geq 1 \). In our case, \( a = 2 \) and \( n = 3 \).

Step 2: Apply the formula to find the time domain function

Substitute \( a = 2 \) and \( n = 3 \) into the formula: \( \mathscr{L}^{-1}\{\frac{1}{(s+2)^3}\} = \frac{t^{3-1}}{(3-1)!} e^{-2t} \).

Step 3: Simplify the expression

Calculate the factorial and power terms: \( \frac{t^{2}}{2!} = \frac{t^2}{2} \). Thus, the inverse Laplace transform is given by \( \frac{t^2}{2} e^{-2t} \).

Key Concepts

Laplace Transform

The Laplace Transform is a powerful tool used extensively in engineering, physics, and mathematics. It converts functions of time into functions of a complex variable, often simplifying the process of solving differential equations.
\( F(s) \) represents the Laplace Transform of a function \( f(t) \), converting a time-domain function into the frequency domain. This transformation can simplify the operations of differentiation and integration into algebraic manipulation.
Key applications include:

• Simplifying the solution of linear differential equations.
• Analyzing linear time-invariant systems, which are common in engineering.
• Converting convolutions in time into simple multiplications in the frequency domain.

Understanding the inverse process allows us to take a frequency-domain expression, often obtained during engineering analysis, back to the original time-domain function for practical implementations.

Differential Equations

Differential equations involve mathematical expressions that relate a function to its derivatives, making them integral in describing various physical phenomena.
With the help of the Laplace Transform, solving differential equations becomes more straightforward as it transforms differential operators into simple algebraic ones.
In these equations:

• Time-dependent behaviors found in systems and processes are modeled.
• Laplace transforms change differentiation into multiplication with \( s \), which dramatically simplifies complex problems.

As in our example, we start with a frequency-domain function, use known Laplace pairs to apply the inverse transform, and return to a time-domain function, tackling practical system descriptions.

Engineering Mathematics

Engineering mathematics encompasses mathematical techniques and problem-solving strategies essential for engineering tasks.
In the field of engineering, mathematics is not just theoretical; it solves real-world problems. Engineers rely heavily on tools like the Laplace Transform for designing circuits, controlling systems, and understanding material behavior.
Core topics include:

• Modeling and analyzing systems through differential equations.
• The application of mathematical transforms, such as Laplace, to manage and resolve complex problems systematically.
• Using mathematics for precision and rigor in designing and manufacturing processes.

By mastering these mathematical tools, engineers enhance problem-solving capabilities and increase efficiency in technology and research development.