Problem 3 - Advanced Engineering Mathematics

Question

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t^{3} e^{-2 t}\right\\} $$

Step-by-Step Solution

Answer

\(\mathscr{L}\{t^3 e^{-2t}\} = \frac{6}{(s+2)^4}\).

Step 1: Write the definition of the Laplace Transform

The Laplace Transform of a function \(f(t)\) is given by the integral \(\mathscr{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt\). In this exercise, we need to find the Laplace Transform of \(f(t) = t^3 e^{-2t}\).

Step 2: Substitute the function into the Laplace Transform definition

Replace \(f(t)\) in the Laplace Transform integral: \[\mathscr{L}\{t^3 e^{-2t}\} = \int_{0}^{\infty} e^{-st} t^3 e^{-2t} \, dt = \int_{0}^{\infty} t^3 e^{-(s+2)t} \, dt.\]

Step 3: Identify the form of the integral

Identify that the integral \(\int_{0}^{\infty} t^n e^{-at} \, dt\) is of the form which leads to \(\frac{n!}{a^{n+1}}\). Here, \(n = 3\) and \(a = s+2\).

Step 4: Apply the formula for the Laplace Transform

Using the integral formula mentioned in Step 3, substitute the values of \(n\) and \(a\):\[\int_{0}^{\infty} t^3 e^{-(s+2)t} \, dt = \frac{3!}{(s+2)^{3+1}} = \frac{6}{(s+2)^4}.\]

Step 5: Write the final result

The Laplace Transform of the given function is:\[\mathscr{L}\{t^3 e^{-2t}\} = \frac{6}{(s+2)^4}.\]

Key Concepts

Integral Transform

Integral transforms are an essential tool used to convert complex functions into a form that can be more easily analyzed. They take a given function and transform it through integration into another function, usually in a different variable. One of the most popular types of integral transforms is the Laplace Transform.

To perform an integral transform, you write the function you want to transform within an integral. Typically, you'll integrate this function over a specific limit, like from zero to infinity. In the case of the Laplace Transform, the formula used is:

• \( \mathscr{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \)

The purpose of this is to take a time-domain function and translate it into a s-domain or frequency domain function. This makes it beneficial for solving differential equations, where dealing in the frequency domain is often easier.

Understanding how integral transforms, especially the Laplace Transform, work provides a strong foundation for analyzing systems dynamically modeled by differential equations. This makes it highly valuable in fields such as engineering, physics, and applied mathematics.

Exponential Function

The exponential function is a fundamental concept in mathematics expressed with a base raised to the power of a variable, most commonly noted as \(e^x\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. This function describes growth and decay processes.

• An important property of exponential functions is their constant rate of growth or decay.
• They serve as solutions to differential equations where the rate of change of a quantity is proportional to the quantity itself.

In the context of the Laplace Transform, exponential functions appear within the integral, as seen in \(e^{-st}\) which helps in transforming the original function. In our original exercise, the function considered is \(f(t) = t^3 e^{-2t}\), where the exponential part is pivotal in shaping the resulting transformed function. This exponential term shifts the spectrum of the function from the time domain to the Laplace space, facilitating simpler manipulation and solution finding.

Gamma Function

The gamma function is a key component in mathematics, often used to extend factorials to non-integer values. It is typically denoted by \( \Gamma(n) \) and satisfies \( \Gamma(n) = (n-1)! \) for positive integers.The gamma function is defined by the integral:

• \( \Gamma(n) = \int_{0}^{\infty} t^{n-1} e^{-t} \, dt \)

This definition is very similar in form to the integral found in the Laplace Transform, particularly when dealing with transformations of polynomials multiplied by an exponential function. In our exercise, the integral \( \int_{0}^{\infty} t^3 e^{-(s+2)t} \, dt \) makes use of the gamma function concept, as it resembles the generalized form \( \int_{0}^{\infty} t^n e^{-at} \, dt \), producing a result based on factorials. Understanding the role of the gamma function can simplify the process of solving Laplace transforms by recognizing recurring integral forms.

Laplace Transform Applications

Laplace Transforms have broad applications across many fields due to their ability to simplify complex differential equations, turning them into straightforward algebraic equations. This makes them very useful in engineering and science for analyzing linear time-invariant systems.Some common applications include:

• Electrical engineering: Transforming circuit equations to solve for current and voltage over time.
• Control systems: Designing systems that respond predictably and stably over time.
• Mechanical engineering: Solving dynamic systems involving motion and heat transfer.

The transformation of time-domain functions into s-domain functions allows for easier manipulation and solving of systems subject to initial conditions. In our example, finding \( \mathscr{L}\{t^3 e^{-2t}\} \) shows how a relatively complex function in time can be translated into a simpler, closed algebraic form, \( \frac{6}{(s+2)^4} \). This translated form provides insight into the system's behavior without the need for cumbersome time-domain calculations.