Question

Find the inverse Laplace transform of the given function. $$ \frac{3}{s^{2}+4} $$

Short answer

Answer: The inverse Laplace transform of the given function is \(f(t) = 3\sin(2t)\).

Step-by-step solution

Identify a Laplace Transform Pair

From the Laplace transform table, we can find a pair of the form: $$ \mathcal{L}\{f(t)\} = \frac{k}{s^{2} + k^2} $$ Where the inverse Laplace transform can be determined as: $$ f(t) = \mathcal{L}^{-1}\{\frac{k}{s^{2} + k^2}\} = k\sin(kt) $$

Match the Given Function with the Laplace Transform Pair

For our given function, we have: $$ \frac{3}{s^{2}+4} $$ By comparing this function with the standard Laplace transform pair, we can identify the constants as: $$ k = \sqrt{4}=2 $$ This means that our function f(t) is scaled by a factor of 3/2.

Determine the Inverse Laplace Transform

Now, we can find the inverse Laplace transform of our given function, using the known inverse Laplace transform of the identified pair: $$ f(t) = \mathcal{L}^{-1}\{\frac{3}{s^{2}+4}\} = \frac{3}{2} (2\sin(2t)) $$ Simplifying the expression: $$ f(t) = 3\sin(2t) $$ The inverse Laplace transform of the given function \(\frac{3}{s^{2}+4}\) is \(f(t) = 3\sin(2t)\).

Key concepts

Laplace Transform Pairs

Understanding Laplace Transform Pairs is essential when finding the inverse Laplace transform of a function. Laplace transform pairs consist of a function in the time domain and its corresponding representation in the Laplace domain. For example, the function \( f(t) = k\sin(kt) \) in the time domain corresponds to \( \frac{k}{s^2 + k^2} \) in the Laplace domain.
This relationship allows us to easily switch between the time and Laplace domains using transformation tables, which list common pairs. By identifying these pairs, we can solve differential equations and analyze systems that are described by these equations.

• Transforms are used to convert differential equations into algebraic equations, which are easier to manipulate.
• Laplace tables help find these transforms quickly and efficiently.

Recognizing these pairs is crucial for engineers and scientists in applications involving control systems and signal processing.

Sine Function

The sine function is a fundamental concept when dealing with waveforms and oscillations. In the context of Laplace transforms, the sine function is often part of the solution for systems experiencing periodic behavior. Specifically, the Laplace transform of a sine function \( \sin(kt) \) is \( \frac{k}{s^2 + k^2} \).
Sine functions represent smooth repetitive oscillations that can describe various physical phenomena, including sound waves and alternating current.

• The amplitude of the sine function determines the peak value of the wave.
• The frequency of the sine function, given by \( k \), determines how fast the oscillation occurs.

Understanding the interplay between the Laplace domain and the time domain representations allows us to accurately capture and solve problems related to waves and oscillations.

Solving Differential Equations

Solving differential equations using Laplace transforms involves transforming the problem into a more manageable algebraic form. This powerful method is especially handy when dealing with linear differential equations with constant coefficients. Once transformed, these equations can be solved more straightforwardly.
The main steps in this process include:

• Applying the Laplace transform to each term in the differential equation.
• Rewriting the initial conditions in the Laplace domain if given.
• Solving the resulting algebraic equation.
• Converting the solution back to the time domain using the inverse Laplace transform.

This technique simplifies the process of solving complex differential equations, making it a favored choice among engineers and mathematicians working with dynamic systems and signals.