Question

Find the inverse Laplace transform. $$F(s)=\frac{2}{s-3}$$

Short answer

Question: Find the inverse Laplace transform of the given function: $$F(s) = \frac{2}{s-3}$$ Answer: The inverse Laplace transform of the given function is $$f(t) = 2e^{3t}$$.

Step-by-step solution

Identify the function's form

We first check if our function $$F(s) = \frac{2}{s-3}$$ is in the standard Laplace transform table. Indeed, it has a similar form to the Laplace transform of $$e^{at}$$ which is given by $$\frac{1}{s-a}$$. In our case, $$a = 3$$ and we have a constant multiplier $$2$$.

Use the inverse Laplace transform property

Now that we know the form, we can directly use the inverse Laplace transform property. If $$L\{e^{at}\} = \frac{1}{s-a}$$, then $$L^{-1}\{\frac{1}{s-a}\} = e^{at}$$. Since we have $$F(s) = 2\cdot\frac{1}{s-3}$$, we can use the property of linearity for inverse Laplace transforms: $$L^{-1}\{2\cdot\frac{1}{s-3}\} = 2\cdot L^{-1}\{\frac{1}{s-3}\}$$

Apply the inverse Laplace transform

Now, applying the inverse Laplace transform to each term, we get: $$2\cdot L^{-1}\{\frac{1}{s-3}\} = 2\cdot e^{3t}$$ So, the inverse Laplace transform of $$F(s) = \frac{2}{s-3}$$ is: $$f(t) = 2e^{3t}$$

Key concepts

Laplace transform table

The Laplace transform table is a crucial tool for engineers and scientists working with differential equations and systems analysis. It's essentially a reference chart that provides a correspondence between functions in the time domain, often denoted as t, and their image functions in the complex frequency domain, denoted as s.

The table includes a list of common functions like polynomials, exponentials, trigonometric, and hyperbolic functions, along with their respective Laplace transforms. For example, one of the most fundamental entries in the table is for the exponential function, where the Laplace transform of e^at is given by \(\frac{1}{s-a}\), assuming convergence of the integral defining the transform for s > a. This relation is straightforward and provides a quick way to switch perspectives from time to frequency domain, or vice versa, which is immensely valuable for solving differential equations.

Linearity of inverse Laplace transforms

Understanding the linearity property of the inverse Laplace transform is vital for simplifying complex transforms into more manageable equations. This property is analogous to the superposition principle used across various fields of physics and engineering.

Stated mathematically, if \(F(s)\) and \(G(s)\) are Laplace transforms of \(f(t)\) and \(g(t)\) respectively, and \(a\) and \(b\) are constants, then the inverse Laplace transform of \(aF(s) + bG(s)\) is \(af(t) + bg(t)\). In practice, this means that we can apply the inverse Laplace transform to each term independently and multiply by their respective constants. The result of this operation will be the sum of the individual inverse transforms, scaled accordingly. This linearity makes the process of finding inverse Laplace transforms from a table much less daunting and is exemplified in the step-by-step solution where the factor of 2 is simply pulled out in front of the inverse transform operation.

Exponential functions

Exponential functions like \(e^{at}\) are omnipresent in the realms of mathematics, physics, and engineering due to their unique properties, especially in growth and decay processes. These functions are characterized by their constant rate of proportional growth or decay.

The function \(e^{at}\), where \(a\) is a constant, represents continuous growth if \(a > 0\) or decay if \(a < 0\). In the context of inverse Laplace transforms, exponential functions are particularly noteworthy since their transforms have a simple form that is easy to recognize and work with. As illustrated in the example problem, the presence of an exponential function's signature in the denominator of \(F(s)\) as \(s-a\) enables us to quickly identify the corresponding time-domain function, leveraging the properties and tables to streamline calculations. This process demystifies the concept of reversing a transformation from the frequency to the time domain, especially for students new to the topic.