Problem 17 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The inverse Laplace transform is \( f(t) = te^{-t} \).

Step 1: Identify the Inverse Laplace Transform

We are given \( \mathscr{L}^{-1}\left\{\frac{s}{(s+1)^{2}}\right\} \). Our task is to find the inverse Laplace transform of this expression, which will result in the original time-domain function \( f(t) \).

Step 2: Check the Formula for Inverse Transform

According to the Laplace transform table, the inverse Laplace transform of \( \frac{s}{s-a}^n \) is given by \( \frac{t^{n-1}}{(n-1)!} e^{at} \). Here \( a = -1 \) and \( n = 2 \).

Step 3: Apply the Inverse Transform Formula

By applying the inverse transform formula to our given expression:\[ f(t) = \mathscr{L}^{-1}\left\{\frac{s}{(s+1)^{2}}\right\} = \frac{t^{2-1}}{(2-1)!}e^{-t} \]Simplifying, we get:\[f(t) = te^{-t}.\]

Key Concepts

Laplace Transform Table

The Laplace Transform Table is an essential tool for solving problems involving transformations between time and frequency domains. It lists common functions and their Laplace transforms, facilitating quick reference. For inverse Laplace transforms, the table helps identify patterns and formulas that allow us to derive time-domain functions from Laplace expressions. When dealing with complex mathematical expressions, this table can streamline the process and simplify calculations.

• Functions are presented alongside their transforms, showing both forward and inverse relationships.
• Essential for identifying standard forms and applying correct transform or inverse formulas.
• By matching expressions to entries in the table, you can determine the original function in the time domain without manually performing lengthy integrations.

Using the Laplace Transform Table effectively requires familiarity with common transforms and an understanding of how to apply given formulas.

Time-Domain Function

The Time-Domain Function represents a function defined in terms of time, usually expressed as \( f(t) \). This function is often the ultimate goal when using inverse transformations, as it reveals the behavior or evolution of a system over time.

• Shows how a system or signal behaves with respect to time.
• Derived from the inverse of a Laplace-transformed function.

In our original problem, we started with a Laplace-transformed equation, trying to uncover its time-domain counterpart. Once we apply the inverse Laplace transform, we translate this into a usable time function, which provides insights into real-world scenarios, like solving differential equations or analyzing dynamic systems.
This specific problem results in \( f(t) = te^{-t} \), a function that can be visualized and interpreted to understand its behavior over time.

Inverse Transform Formula

The Inverse Transform Formula turns a Laplace-transformed function back into its equivalent function in the time domain. This process involves recognizing structural patterns within the Laplace expression and using predefined formulas or techniques to reverse the transform.

• Important for converting frequency domain solutions back into time domain insights.
• Relies on known formulas, often present in the Laplace Transform Table.
• Typically involves recognition of the form: \( \frac{s}{(s-a)^n} \).

For our given example, we needed to transform \( \mathscr{L}^{-1}\left\{\frac{s}{(s+1)^{2}}\right\} \) into \( f(t) \). By identifying that \( a = -1 \) and \( n = 2 \), and applying the formula \( \frac{t^{n-1}}{(n-1)!} e^{at} \), we derived \( f(t) = te^{-t} \). This inverse process gives a clear view of time-related phenomena from their complex expressions in the Laplace domain.