Problem 18 - Advanced Engineering Mathematics

Question

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{5 s}{(s-2)^{2}}\right\\} $$

Step-by-Step Solution

Answer

The inverse Laplace transform is \( 5t e^{2t} \).

Step 1: Identify the Inverse Laplace Transform Form

The given expression is \( \mathscr{L}^{-1}\left\{\frac{5s}{(s-2)^2}\right\} \). This expression needs to be matched to known inverse Laplace transform forms. Here, it resembles a function of the form \( \mathscr{L}^{-1}\{\frac{s-a}{(s-a)^2}\} \).

Step 2: Use Shifting Theorem

Recognize that the expression \( \frac{5s}{(s-2)^2} \) suggests a first order derivative of an exponential function due to the \( (s-a) \) shift factor. This is aligned with the derivative of \( e^{at} \) in the time domain. Here, \( a = 2 \).

Step 3: Apply the Derivative Property

The inverse Laplace transform of \( \frac{s-a}{(s-a)^2} \) is \( t e^{at} \). Therefore, for \( a = 2 \), the transform becomes \( t e^{2t} \).

Step 4: Adjust for Coefficient 5

The presence of the coefficient 5 in the numerator means that the expression should be multiplied by 5. Thus, the transform becomes \( 5t e^{2t} \).

Key Concepts

Laplace transform properties

Laplace transforms are powerful tools for solving differential equations and analyzing linear time-invariant systems. The core idea is to transform a function of time, such as \(f(t)\), into a function of a complex variable, \(s\), denoted as \(F(s)\). This process simplifies calculations, especially when dealing with linear systems.

The main properties of the Laplace transform that make it so effective include:

• Linearity: For functions \(f(t)\) and \(g(t)\) and constants \(a\) and \(b\), the transform \(\mathscr{L}\{af(t) + bg(t)\} = aF(s) + bG(s)\).
• Time Shifting: If \(\mathscr{L}\{f(t)\} = F(s)\), then \(\mathscr{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)\), where \(u(t)\) is the unit step function.
• Frequency Shifting: The transform of \(e^{at}f(t)\) is \(F(s-a)\), useful for including exponential growth or decay effects in the analysis.

Understanding these properties is essential because they provide the tools needed to handle various transformations, especially when working with the inverse Laplace transform, as shown in the original exercise.

shifting theorem

The shifting theorem is one of the most crucial concepts in the context of Laplace transforms. It allows one to account for the effects of exponential functions, like \(e^{at}\), in the time domain.

For the given problem, the shifting theorem helps in translating the complex function \( \frac{s-a}{(s-a)^{2}} \) in the \(s\)-domain back into the time domain. Specifically, if you have a Laplace transform in the form of \(F(s-a)\), it signals the presence of an exponential function \(e^{at}\) in \(f(t)\).

Using this theorem, when we look at \( \frac{5s}{(s-2)^{2}} \), the fact that we have \((s-a)\) with \(a = 2\) gives us the hint that the corresponding time function involves \(e^{2t}\). This shifting property simplifies handling terms involving exponential growth or decay, making it easier to revert back to the time domain while dealing with periodic or decaying functions.

derivative property

The derivative property of the Laplace transform involves the relationship between differentiation in the time domain and algebraic manipulation in the \(s\)-domain. This property is valuable when dealing with differential equations.

When applying the inverse Laplace transform to expressions involving derivatives, the property becomes essential. In our exercise, the expression \( \frac{5s}{(s-2)^{2}} \) suggests the use of the derivative property. Here, \( \frac{s-a}{(s-a)^{2}} \) resembles the derivative of a transformed function related to \(t \cdot f(t)\).

• Specifically, this indicates that the inverse transform relates to the function multiplied by time \(t\).
• This relation is crucial because it tells us that \( \frac{s-a}{(s-a)^{2}} \) in the \(s\)-domain corresponds to \(t \cdot e^{at}\) in the time domain. When \(a = 2\), this becomes \(t \cdot e^{2t}\).

Understanding this property allows one to transition from the transform space back to physical time-space, discovering how a system behaves in reality. The concept is integral in constructing solutions to problems involving dynamic systems and change over time.