Question

Use the result of Problem 28 to find the Laplace transform of the given function; \(a\) and \(b\) are real numbers and \(n\) is a positive integer. $$ t^{n} $$

Short answer

Answer: The Laplace transform of the function \(t^n\) is \(\mathcal{L}\{t^n\}(s) = \frac{n!}{s^{n+1}}\).

Step-by-step solution

Recall the formula for the Laplace transform of a power function

The Laplace transform of a power function, \(t^n\), where \(n\) is a positive integer, is given by the formula: $$ \mathcal{L}\{t^n\}(s) = \frac{n!}{s^{n+1}} $$

Apply the formula to the given function

We are given the function \(t^n\). To find its Laplace transform, we can directly use the formula mentioned in Step 1. Substitute \(n\) in the formula with the given value of \(n\): $$ \mathcal{L}\{t^n\}(s) = \frac{n!}{s^{n+1}} $$

Find the Laplace transform of \(t^n\)

From Step 2, we have the Laplace transform of the given function \(t^n\) as: $$ \mathcal{L}\{t^n\}(s) = \frac{n!}{s^{n+1}} $$

Result

The Laplace transform of the function \(t^n\) is: $$ \mathcal{L}\{t^n\}(s) = \frac{n!}{s^{n+1}} $$

Key concepts

Laplace transform of power functions

The Laplace transform is a vital tool used in engineering and physics to simplify the analysis of linear systems, particularly when dealing with differential equations. Among various functions, the Laplace transform of power functions is just one example of its broad applicability. By definition, when we have a power function in the form of t raised to the power of a positive integer n, the Laplace transform is utilized to convert this function from the time domain to the complex frequency domain.

In practical terms, to find the Laplace transform of a power function, one would use the formula
\[\mathcal{L}\{t^n\}(s) = \frac{n!}{s^{n+1}}\]
where n! represents the factorial of n and s is a complex number representing frequency. Understanding this transformation is crucial because it simplifies the process of solving differential equations that model physical systems, such as electrical circuits or mechanical oscillators.

Students can use this formula directly to find the Laplace transform for any given power of t where the exponent is a positive integer; this standardized process facilitates quicker and more efficient problem-solving in various fields of study.

power function

At the heart of the Laplace transform of power functions is the power function itself. A power function is a simple algebraic function that can be described as t^n, where t is a real number and represents the variable (commonly time in physical problems), and n is a positive integer serving as the exponent.

Power functions have diverse applications in mathematics and the sciences; they're used to model an array of phenomena from the growth of populations to the physics of falling objects. It's the simplicity of these functions that makes them so powerful (pun intended) and foundational. They are easy to differentiate and integrate, which is why they appear frequently in problems related to rates of change and areas under curves in calculus.

In the context of Laplace transforms, the significance of power functions is amplified since they become the focus of transformation when looking to solve differential equations governing systems dynamics, allowing for an approach to complex problems using simple algebraic methods.

positive integer

Understanding the nature of the positive integer within the formula for the Laplace transform of power functions is key to its correct application. Simply put, a positive integer is a whole number greater than zero, which in this context is used as the exponent in the power function t^n.

It's important to reiterate that the Laplace transform only works with positive integers in this formula, as it ensures the resulting transform is a proper rational function that can be manipulated algebraically. When dealing with exponentials in differential equations, positive integers keep the mathematics tidy and manageable. Moreover, they are integral to the factorial operation (n!) in the transform's formula, which is only defined for non-negative integers. In more advanced mathematics, the concept of factorial can be extended beyond the positive integers, but within the scope of the Laplace transform for power functions, working within this limitation is essential for maintaining the integrity of the solutions.

In summary, the importance of the positive integer in the Laplace transform equation cannot be overstated; it is a critical component that ensures the equation is applicable and the solution is tangible for physical and engineering problems.