Question

Find the inverse Laplace transform. $$F(s)=\frac{1}{(s+1)^{3}}$$

Short answer

Question: Find the inverse Laplace transform of the function $$F(s) = \frac{1}{(s+1)^{3}}$$. Answer: The inverse Laplace transform of the given function is $$f(t) = t^2 e^{-t} u(t)$$.

Step-by-step solution

Identify the basic Laplace transform

To find the inverse Laplace transform of the given function, we first need to identify what basic Laplace transform this function is related to. The function resembles the Laplace transform of the 2nd derivative of the unit step function, which is: $$L\{t^n e^{-at} u(t)\} = \frac{n!}{(s+a)^{n+1}}$$ Comparing it with the given function, we can rewrite the given function in this form: $$F(s) = \frac{1}{(s+1)^{3}} = \frac{2!}{(s+1)^{3}}$$

Apply the inverse Laplace transform formula

Now we apply the inverse Laplace transform formula to the function: $$f(t) = \mathcal{L}^{-1}\{F(s)\} = \mathcal{L}^{-1}\{\frac{2!}{(s+1)^{3}}\}$$ Using the known inverse Laplace transform for the derivative of the unit step function: $$f(t) = t^2 e^{-t} u(t)$$

Simplify the result

The inverse Laplace transform of the given function is: $$f(t) = t^2 e^{-t} u(t)$$ Where \(u(t)\) is the unit step function. This is the final result.

Key concepts

Laplace Transform

The Laplace Transform is a powerful mathematical tool used to transform complex differential equations into simpler algebraic equations. This makes it a popular technique in engineering and physics.
The transform takes a time-domain function and converts it into a complex frequency-domain function. The Laplace Transform is defined as:
\[L\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\]
This transformation is particularly useful for solving linear differential equations, system dynamics, and control systems. By mapping the function into the s-domain, it becomes easier to analyze stability, conduct system analysis, or synthesis.
The process involves taking into account initial conditions that govern the system's response. These are encapsulated in the frequency domain representation, which can be manipulated algebraically to arrive at a solution. Hence, the intricate time-based function f(t) is expressed in a much simpler form as F(s).

Unit Step Function

The Unit Step Function, represented as \(u(t)\), is a fundamental piecewise function characterized by its step-like behavior.
This function is defined as:
\[u(t) = \begin{cases} 0, & t < 0 \ 1, & t \geq 0 \end{cases}\]
It serves as a basic building block for more complex functions in signal processing and control systems.
In the context of Laplace Transforms, the unit step function allows for the representation and manipulation of signals that start at a particular time -- typically at zero. By multiplying any time function by \(u(t)\), you effectively 'turn it on' at \(t = 0\). This ability is crucial when dealing with real-world systems that only operate for \(t \geq 0\).
It ties closely with Laplace Transforms as it provides boundaries and conditions that define the function's starting point, making it easier to manage and determine inverse transforms.

Differential Equations

Differential Equations are mathematical equations that involve derivatives of a function. They are used to describe various phenomena such as motion, electricity, heat, fluid dynamics, and more.
The two main types are ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of one variable, while PDEs involve functions of multiple variables.
Solving these equations analytically can be challenging, but with tools like the Laplace Transform, it becomes more feasible. The transform allows one to handle linear ODEs by converting differential operations into multiplication, which simplifies the complexity involved.
This is incredibly useful in engineering, where systems are often modeled as sets of differential equations that describe reactions over time. The transformed equations can then be easily manipulated to solve for the behavior of the system.

Mathematical Analysis

Mathematical Analysis is a branch of mathematics that deals with limits, integration, differentiation, and continuity. It provides the theoretical foundation for many mathematical constructs, including the Laplace Transform and differential equations.
Through analysis, one can explore the convergence of functions, the behavior of dynamic systems, or perform complex integrals and series expansions. This investigation and rigorous assessment lead to improved understanding and precision in mathematical modeling and problem-solving.
In the case of thermodynamics, electronics, or other applications, mathematical analysis helps detail the properties of the system.
For instance, the process of finding an inverse Laplace Transform relies heavily on analysis techniques to ensure that each operational step, manipulation, or simplification follows from a well-founded mathematical rationale.
Each transformation in the frequency domain can be analyzed for its implications in the time domain, ensuring that models act predictably and correctly mirror physical realities.