Problem 6 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

\( \frac{2e^{2-s}}{(s-2)^3} \)

Step 1: Recognize the Laplace Transform of a Shifted Function

The Laplace transform of a function multiplied by an exponential term, such as \( e^{at} \), is a shifting in the \( s \)-domain. The formula to use here is \( \mathscr{L}\{ e^{at}f(t) \} = F(s-a) \). We identify \( a = 2 \) and \( f(t) = (t-1)^2 \).

Step 2: Find the Laplace Transform of \( (t-1)^2 \)

We need to find the Laplace transform of \( (t-1)^2 \). Using the fact that \((t-a)^n\) transforms to \( e^{-as}\frac{n!}{s^{n+1}} \) with \( a = 1 \) and \( n = 2 \), the Laplace transform \( \mathscr{L}\{(t-1)^2\} \) is \( e^{-s}\frac{2}{s^3} \).

Step 3: Apply the Shifting Property

Now apply the shifting property: replace \( s \) by \( s-2 \) in the transform of \( (t-1)^2 \). The transform becomes \( e^{-(s-2)}\frac{2}{(s-2)^3} \), which simplifies to \( \frac{2}{(s-2)^3}e^{-s+2} \).

Step 4: Simplify the Exponentials

Simplify the expression \( e^{-s+2} \) by combining exponents: \( e^{-s}e^{2} \). Therefore, the Laplace transform is \( 2e^{2}\frac{e^{-s}}{(s-2)^3} \), or \( \frac{2e^{2-s}}{(s-2)^3} \).

Key Concepts

Laplace transform of shifted functions

The Laplace transform plays a crucial role in transforming complex differential equations into simpler algebraic problems. One fascinating scenario occurs when a function is multiplied by an exponential term, which results in a shift in the frequency variable, denoted by \( s \).
When you have a function like \( e^{at}f(t) \), its Laplace transform is not straightforwardly \( F(s) \), but rather involves a shift. The formula for this is given as \( \mathscr{L}\{ e^{at}f(t) \} = F(s-a) \).

• \( a \) is the constant that multiplies \( t \) inside the exponent.
• \( f(t) \) is the function being transformed.

This shifting property is designed to adjust for the exponential increase or decrease in the function's behavior due to \( e^{at} \). Understanding this property is essential to correctly applying the Laplace transform to different functional forms.

exponential shifting property

When a function is altered by an exponential, it impacts the original function's Laplace transform. This phenomenon is known as the exponential shifting property.
The core idea is that the multiplication by \( e^{at} \) results in a shift by \( a \) units in the \( s \)-domain. Consequently, if you know the Laplace transform \( F(s) \) of a function \( f(t) \), the transform of \( e^{at}f(t) \) becomes \( F(s-a) \).

• This shift reflects how the exponential factor impacts the frequency characteristics of a function.
• It is a crucial tool for analyzing systems, especially in engineering fields.

The property allows engineers and mathematicians to handle complex transformations with ease by focusing on simple algebraic manipulations in the \( s \)-domain rather than grappling with the original differential equation.

steps for solving Laplace transforms

Applying the Laplace transform involves a structured approach to go from a time-domain expression to its frequency domain. Let's breakdown the steps:

• **Identify the components**: Isolate the function \( f(t) \) and any exponential factors \( e^{at} \).
• **Transform the base function**: Calculate the Laplace transform of the base function \( f(t) \) if known, using a table or known formulas. In the example, \( (t-1)^2 \) transforms to \( e^{-s}\frac{2}{s^3} \).
• **Apply the shift**: Use the exponential shifting property. Replace \( s \) with \( s-a \) in the transform. For \( a = 2 \), the transform becomes \( \frac{2}{(s-2)^3}e^{-s+2} \).
• **Simplify**: Combine terms and simplify exponentials, resulting in \( \frac{2e^{2-s}}{(s-2)^3} \).

These steps emphasize a clear sequence that aids in confidently solving these problems, making them less intimidating and more approachable for beginners.