Laplace Transform Of 1/(s(s^2+1)) Explained

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Navigating the realm of Laplace transforms often presents intriguing challenges, and determining the Laplace transform of 1/(s(s^2+1)) is a classic example. This article delves into a step-by-step solution, providing a clear understanding of the underlying concepts and techniques. We will explore the application of partial fraction decomposition and the inverse Laplace transform to arrive at the correct answer, which is 1 - cos(t). This exploration will not only equip you with the solution but also enhance your understanding of Laplace transforms and their applications in various fields, including engineering and physics.

Understanding Laplace Transforms

Before we dive into the specifics of this problem, let's establish a solid foundation in Laplace transforms. The Laplace transform is a powerful mathematical tool that transforms a function of time, denoted as f(t), into a function of complex frequency, denoted as F(s). This transformation is particularly useful in solving linear differential equations, as it converts differential equations into algebraic equations, which are often easier to solve. The Laplace transform is defined by the following integral:

L{f(t)} = F(s) = ∫0∞ e^(-st) f(t) dt

Where:

  • L denotes the Laplace transform operator.
  • f(t) is the function of time.
  • F(s) is the Laplace transform of f(t).
  • s is a complex frequency variable (s = σ + jω, where σ and ω are real numbers and j is the imaginary unit).

The inverse Laplace transform, denoted as L^(-1){F(s)}, converts a function in the complex frequency domain back to a function in the time domain. This is crucial for obtaining the solution to the original problem in terms of time.

The Laplace transform finds widespread application in electrical engineering, control systems, signal processing, and various other domains. Its ability to simplify complex differential equations makes it an indispensable tool for analyzing dynamic systems and designing control algorithms. For instance, in circuit analysis, the Laplace transform can be used to analyze the transient behavior of circuits and determine the stability of systems. In control systems, it aids in designing controllers that ensure the desired system response and stability.

Decomposing the Fraction: Partial Fraction Decomposition

To find the Laplace transform of 1/(s(s^2+1)), the first crucial step is to employ partial fraction decomposition. This technique allows us to break down a complex rational function into simpler fractions, each of which has a known inverse Laplace transform. The given function, 1/(s(s^2+1)), can be decomposed into the following form:

1/(s(s^2+1)) = A/s + (Bs + C)/(s^2+1)

Where A, B, and C are constants that we need to determine. To find these constants, we multiply both sides of the equation by the common denominator, s(s^2+1), which gives us:

1 = A(s^2+1) + (Bs + C)s

Expanding the right side of the equation, we get:

1 = As^2 + A + Bs^2 + Cs

Now, we group the terms with the same powers of s:

1 = (A + B)s^2 + Cs + A

By equating the coefficients of the corresponding powers of s on both sides of the equation, we obtain a system of linear equations:

  • A + B = 0 (coefficient of s^2)
  • C = 0 (coefficient of s)
  • A = 1 (constant term)

From these equations, we can easily solve for A, B, and C. We find that A = 1, C = 0, and B = -1. Substituting these values back into the partial fraction decomposition, we get:

1/(s(s^2+1)) = 1/s - s/(s^2+1)

This decomposition is a key step in simplifying the problem, as it allows us to work with simpler functions whose inverse Laplace transforms are readily available. The ability to decompose complex fractions is a fundamental skill in Laplace transform analysis, and it is used extensively in various engineering and mathematical applications. Partial fraction decomposition is not only applicable in Laplace transforms but also in integral calculus and other areas of mathematics, making it a versatile tool in your mathematical arsenal.

Applying the Inverse Laplace Transform

With the function 1/(s(s^2+1)) decomposed into simpler fractions, the next step is to apply the inverse Laplace transform to each term. This process transforms the function from the complex frequency domain back to the time domain. We have the decomposed form:

1/(s(s^2+1)) = 1/s - s/(s^2+1)

Now, we apply the inverse Laplace transform to both sides of the equation:

L(-1){1/(s(s2+1))} = L^(-1){1/s} - L(-1){s/(s2+1)}

We can now use standard Laplace transform pairs to find the inverse Laplace transform of each term. Recall the following common Laplace transform pairs:

  • L^(-1){1/s} = 1
  • L(-1){s/(s2+a^2)} = cos(at)

In our case, a = 1. Applying these pairs, we get:

L^(-1){1/s} = 1

L(-1){s/(s2+1)} = cos(t)

Substituting these results back into the equation, we obtain:

L(-1){1/(s(s2+1))} = 1 - cos(t)

Therefore, the Laplace transform of 1/(s(s^2+1)) is 1 - cos(t). This result matches option (a) in the given choices. The process of applying the inverse Laplace transform is a critical step in solving differential equations using the Laplace transform method. It allows us to obtain the solution in the time domain, which is often the desired form in practical applications. Understanding the common Laplace transform pairs and being able to apply them correctly is essential for mastering Laplace transform techniques.

Conclusion: Mastering Laplace Transforms

In conclusion, we have successfully determined the Laplace transform of 1/(s(s^2+1)) by employing partial fraction decomposition and applying the inverse Laplace transform. The correct answer is 1 - cos(t). This exercise highlights the importance of understanding fundamental concepts and techniques in Laplace transform analysis.

The Laplace transform is a versatile tool with applications in various fields, including electrical engineering, control systems, and signal processing. By mastering the techniques discussed in this article, you will be well-equipped to tackle a wide range of problems involving dynamic systems and differential equations. This includes the ability to decompose complex fractions using partial fraction decomposition, identify and apply standard Laplace transform pairs, and accurately perform the inverse Laplace transform.

Furthermore, the problem-solving approach demonstrated here can be applied to other complex functions. The key is to break down the problem into manageable steps, apply the appropriate techniques, and carefully track each step to arrive at the correct solution. With practice and a solid understanding of the underlying principles, you can confidently navigate the complexities of Laplace transforms and leverage their power in your field of study or work. The journey of mastering Laplace transforms is continuous, and each solved problem adds to your expertise and confidence.

By understanding the step-by-step process and the underlying principles, you can confidently tackle similar problems and further expand your knowledge in the fascinating world of Laplace transforms. This mastery will undoubtedly prove valuable in your academic and professional pursuits.