Problem 38 - Differential Equations and Linear Algebra

Question

Determine \(L^{-1}[F]\). $$F(s)=\frac{s}{(s+1)^{2}+4}$$.

Step-by-Step Solution

Answer

The inverse Laplace transform of the given function is \(L^{-1}[F] = e^{-t}\frac{2}{\pi}\sin(2t)\).

Step 1: Identify the function format

The given function has the form: \[F(s) = \frac{s}{(s+1)^{2}+4}\] The function can be rewritten as: \[F(s) = \frac{s}{(s+1)^{2}+2^2}\]

Step 2: Identify a and b

From the format, we can observe that: \[a = -1\text{ and }b = 2\]

Step 3: Use the formula for the inverse Laplace transform

The inverse Laplace transform for a function of the form \(\frac{s-a}{(s-a)^{2}+b^2}\) is given by \(e^{at}\frac{b}{\pi} \sin(bt)\). Applying this formula to our function, we get: \(L^{-1}[F] = e^{-1t}\frac{2}{\pi}\sin(2t)\)

Step 4: Final Result

The inverse Laplace transform of the given function is: \[L^{-1}[F] = e^{-t}\frac{2}{\pi}\sin(2t)\]

Key Concepts

Laplace Transforms

Laplace Transforms are a mathematical technique that is particularly useful for solving differential equations. The Laplace Transform converts functions from the time domain into the frequency domain, making complex operations more manageable. For example, integration and differentiation in the time domain become multiplication and division by polynomials in the frequency domain. This transformation is especially powerful when dealing with linear time-invariant systems. To perform a Laplace transform, we have a standard formula: \[\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} f(t)e^{-st}dt\] When we talk about inverse Laplace Transforms, we're looking to recover the original time-domain function from its frequency-domain representation. In this exercise, we saw an example of using the inverse Laplace transform to revert a given function:

• Analyze the given function format.
• Identify constants for transformation.
• Apply the inverse formula to obtain the function in time domain.

This method helps manage various engineering problems by simplifying differential equations.

Differential Equations

Differential Equations are equations involving derivatives, and their behavior can often be simplified using Laplace Transforms. Such equations model how quantities change over time and are fundamental in fields such as physics, engineering, and economics. Solving a differential equation typically involves finding a function or set of functions that satisfy the equation. Transforming these equations to the s-domain using Laplace can simplify the solution process. Here is why Laplace Transforms are handy:

• They turn differential equations into algebraic equations, which are generally easier to solve.
• Initial condition handling becomes more straightforward through addition and constants in the equation.
• For many problems, once the solution in the s-domain is found, inverse transformations bring it back to the time-domain.

In the provided exercise, we implicitly use transformations to work with functions easier, allowing systematic solutions to these otherwise difficult equations.

Complex Analysis

Complex Analysis involves the study of complex numbers and their functions. It's a crucial area of mathematics that provides insights into many practical and theoretical problems. The key relation between complex analysis and Laplace transforms lies in their use of complex numbers and functions. Laplace Transforms often involve complex functions, particularly when employed with contour integration – a concept from complex analysis. In this exercise, Laplace involves terms of the form \((s-a)^2 + b^2\), linking it directly to complex numbers. Here's how complex analysis fits into the picture:

• It helps in understanding the poles and zeros of transformed functions.
• It aids in performing contour integrations, a method to evaluate complex integrals that carry over to inverse Laplace transforms.
• It simplifies the analysis of time and frequency domain relationships in engineering.

Complex analysis is a powerful backdrop that supports the transformative processes used in solving differential equations and in interpreting inverse Laplace Transforms.