Problem 11 - Advanced Engineering Mathematics

Question

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s+2)^{3}}\right\\} $$

Step-by-Step Solution

Answer

The inverse Laplace transform is \( f(t) = \frac{t^2}{2} e^{-2t} \).

Step 1: Identify the Problem Type

The given problem is to find the inverse Laplace transform of \( \frac{1}{(s+2)^3} \).

Step 2: Recognize the Transform Form

The expression \( \frac{1}{(s+a)^n} \) corresponds to a known inverse Laplace transform. Specifically, the inverse transform \( \mathscr{L}^{-1} \) of \( \frac{1}{(s+a)^n} \) is \( \frac{t^{n-1}}{(n-1)!}e^{-at} \) for \( t \geq 0 \).

Step 3: Match the Values with the Formula

For the expression \( \frac{1}{(s+2)^3} \), we identify \( a = 2 \) and \( n = 3 \).

Step 4: Apply the Inverse Laplace Transform Formula

Using the formula from Step 2, substitute \( a = 2 \) and \( n = 3 \):\[ f(t) = \frac{t^{3-1}}{(3-1)!}e^{-2t} = \frac{t^2}{2}e^{-2t} \].

Step 5: Simplify the Expression

Therefore, the inverse Laplace transform of \( \frac{1}{(s+2)^3} \) is given by:\[ f(t) = \frac{t^2}{2}e^{-2t} \].

Key Concepts

Laplace Transform

The Laplace Transform is a widely used integral transform in mathematics with applications in engineering and physics. It converts functions from the time domain, typically denoted as \(f(t)\), into the frequency domain, represented as \(F(s)\). This transform is particularly useful for solving linear differential equations.

Laplace Transforms work by integrating a given function multiplied by an exponential decay over time. This process creates a new function of a complex variable \(s\). The formula for a Laplace Transform is given by:\[F(s) = \mathscr{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt.\]

• Simplifies analysis of systems and solves differential equations by turning operations like differentiation into algebraic expressions.
• Helps in modeling and control of dynamic systems.
• Turns initial value problems into simpler algebraic problems.

Through algebraic manipulation, this method opens doors to computationally managing otherwise complex problems.

Partial Fractions

Partial Fractions is a technique used to break rational expressions into simpler fractions that are easier to work with. Particularly useful in integration and taking inverse Laplace Transforms, this method is instrumental in manipulating expressions to a form suitable for known inverse transforms.

The process typically involves:

• Expressing the function as a sum of simpler fractions.
• Solving for unknown coefficients in these fractions.
• Simplifying complex rational expressions to manageable pieces.

Consider a function resembling \( \frac{1}{(s+2)^3} \). Although not requiring partial fraction decomposition due to its polynomial order and form, knowing this technique is essential for more complex denominators, turning them into forms we can inverse-transform easily. This method is vital when dealing with polynomials in Laplace Transform problems.

Differential Equations

Differential Equations describe relationships involving rates of change and are vital in modeling real-world phenomena such as motion, electricity, heat flow, and more. The Laplace Transform is a powerful tool in solving these equations by transforming differential operations into algebraic ones.

When transformed into the frequency domain, a differential equation with time-derivatives can be tackled as an algebraic equation. Consider:\[ \mathscr{L}\{f'(t)\} = sF(s) - f(0)\]

• This conversion provides solutions without directly solving the differential form.
• Commonly used in systems control and engineering.
• Solves for arbitrary initial conditions easily.

In combining Laplace with inverse transformations, engineers and mathematicians can determine time-domain solutions of these important equations effortlessly.

Time-Domain Solutions

Time-Domain Solutions involve finding the behavior of a system in its original domain—typically time—making inverse Laplace Transforms crucial. When converting a problem back to the time domain, you decode the behavior of a dynamic system.

For instance, in the problem provided, the inverse Laplace Transform of \( \frac{1}{(s+2)^3} \) yields \( f(t) = \frac{t^2}{2}e^{-2t} \). This shows:

• The time-dependent behavior of the original differential system.
• Help in determining system response over time, such as resonance in circuits or displacement in mechanical systems.
• Captures both transient and steady-state behaviors.

By mastering inverse transforms, you open insights into how systems evolve over time. Thus, it supports the design and analysis of structures or processes, allowing them to be managed and optimized effectively.