Question

\(L^{-1}\left[\frac{x}{(2 s+3)^{2}}\right]=\ldots\)

Short answer

The inverse Laplace transform is \( x \cdot t \cdot e^{-\frac{3}{2}t} \).

Step-by-step solution

Identify the form

Notice that the function given is \( X(s) = \frac{x}{(2s+3)^2} \). This can be related to a standard Laplace transform form. We recognize a derivative form \( L[f(t)] = \frac{n!}{(s-a)^{n+1}} \). Here, \( a = -\frac{3}{2} \) and \( n = 1 \).

Use Laplace inverse transformation formula

According to the Laplace inverse formula for derivatives, if \( X(s) = \frac{1}{(s-a)^2} \), then \( L^{-1}[X(s)] = t e^{at} \). Adjust this formula to fit the given function where \( a = -\frac{3}{2} \), and solve for the inverse.

Apply the constant multiplier

In this problem, there is a constant multiplier outside of the transform, \( x \). This means the inverse Laplace transform should be multiplied by this constant, resulting in \( L^{-1}[x X(s)] = x L^{-1}[X(s)] \).

Plug into formula

Replace \( a \) in the formula with \( -\frac{3}{2} \) and apply the constant \( x \) multiplier: \[ L^{-1}\left[ \frac{x}{(2s+3)^2} \right] = x \cdot t \cdot e^{-\frac{3}{2}t}. \]

Key concepts

Laplace Transform

The Laplace Transform is a powerful tool used in mathematics and engineering to simplify the solution of differential equations. Essentially, it transforms complex time-domain functions into simpler s-domain functions. This works beautifully with linear systems, where it converts differential equations into algebraic equations, making them easier to manipulate. The key idea behind the Laplace Transform is taking a time-based function, typically denoted as \( f(t) \), and transforming it into \( F(s) \) in the frequency domain using the formula:

• \( L[f(t)] = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt \)

Between these domains, the Laplace Transform creates a bridge where complex operations in the time domain become elementary actions like multiplication and division in the frequency domain. This transformation immensely facilitates solving differential equations arising in various fields, including engineering, physics, and beyond. Furthermore, the inverse Laplace Transform allows us to transform the function back to the time domain, facilitating the analysis of system dynamics.

Differential Equations

Differential equations describe relationships between functions and their derivatives, expressing how a change in one variable affects another. These equations are fundamental in modeling natural phenomena, engineering systems, and many scientific fields. A differential equation takes the form of equations like \( \frac{dy}{dt} = ky \), where the solution requires integrating with respect to the dependent variable.

• They are categorized into ordinary differential equations (ODEs) and partial differential equations (PDEs) based on the type and number of variables.
• Solving these equations can establish the behavior of different systems over time, ranging from predicting the growth of populations to designing control systems in mechanical engineering.

Laplace Transforms simplify this task by turning these differential equations into algebraic ones, converting derivatives into products involving \( s \), and allowing straightforward algebraic manipulation. Consequently, applying the Laplace Transform can unveil solutions that might be intricate or otherwise hard to derive using standard integration techniques.

Engineering Mathematics

In engineering, mathematics is not just theoretical; it is practical. From calculating stresses in materials to designing electronic filters, mathematical tools are foundational. Engineering mathematics often grapples with complex real-world systems through simplified models, using concepts like the Laplace Transform.

• Engineers apply these transforms to model and analyze electrical circuits, vibrations in structures, and control systems, turning complex differential equations into simpler algebraic formats.
• They rely on this transformation to predict system behavior, optimize designs, and ensure stability and efficiency in engineering projects.

Using the inverse Laplace Transform in engineering allows professionals to translate their theoretical models back into real-time predictions, providing insight into system responses. This ability is crucial for developing accurate and functional engineering solutions, enabling engineers to solve differential equations that arise from their designs efficiently.