Question

Find the inverse Laplace transform of the given function. $$ \frac{2 s+2}{s^{2}+2 s+5} $$

Short answer

Answer: The inverse Laplace transform of the given function is \(e^{-t}\left(\frac{2}{5}\cos(2t)+\frac{1}{4}\sin(2t)\right)\).

Step-by-step solution

Perform Partial Fraction Decomposition

For this step, we begin by performing partial fraction decomposition on the given function. $$ \frac{2s+2}{s^2+2s+5} = \frac{A}{s+\alpha} + \frac{Bs + C}{s^2+\beta s+\gamma} $$ However, we can see that the denominator is already in a quadratic form that cannot be further factorized. Thus, partial fraction decomposition will be in this form: $$ \frac{2s + 2}{s^2 + 2s + 5} = \frac{As + B}{s^2 + 2s + 5} $$ Now we need to solve for A and B.

Solve for A and B

Multiply both sides of the equation by the denominator \((s^2 + 2s + 5)\): $$ 2s + 2 = (As + B)(s^2 + 2s + 5). $$ Expanding the equation, we get: $$ 2s + 2 = As^2 + 2As + 5As + 5B $$ By comparing coefficients of similar terms, we get: $$ A + 2A + 5A = 2 \Rightarrow A = \frac{2}{8} = \frac{1}{4} $$ Furthermore, we have: $$ 5B = 2 \Rightarrow B = \frac{2}{5} $$ Now that we have A and B, we can rewrite our function:

Rewrite the function with A and B

With the values of A and B, rewrite the function as: $$ \frac{2s + 2}{s^2 + 2s + 5} = \frac{\frac{1}{4}s + \frac{2}{5}}{s^2 + 2s + 5} $$

Find the Inverse Laplace Transform

Check Laplace transform table to find inverse Laplace transforms for this type of function. We have the form: $$ \mathcal{L}^{-1}\{ \frac{as+b}{s^2+2asw+ws^2} \} = e^{-at} (b\cos(wt)+a\sin(wt)) $$ So in our case, \(a=1\), \(b=\frac{2}{5}\) and \(w=2\). Using the table result, apply inverse Laplace transform to get the result: $$ \mathcal{L}^{-1}\{ \frac{\frac{1}{4}s + \frac{2}{5}}{s^2 + 2s + 5} \} = e^{-t}\left(\frac{2}{5}\cos(2t)+\frac{1}{4}\sin(2t)\right) $$

Final Answer

The final answer for the inverse Laplace transform of the given function is: $$ e^{-t}\left(\frac{2}{5}\cos(2t)+\frac{1}{4}\sin(2t)\right) $$

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify complex rational expressions, especially when dealing with Laplace transforms. The goal is to break down a fraction with a polynomial in the denominator into a sum of simpler fractions, where the denominators are factors of the original denominator. This can make it easier to perform operations like finding the inverse Laplace transform.

Application in Inverse Laplace Transforms

When you have a rational expression where the degree of the numerator is less than the degree of the denominator, partial fraction decomposition can be particularly useful. Once decomposed, each term of the broken down expression often has a straightforward inverse Laplace transform, based on standard forms found in Laplace transform tables. Remember, it is critical to first determine whether the denominator can be factorized further. As shown in the step-by-step solution for our exercise, when it cannot, like with an irreducible quadratic, we express the numerator as a linear combination of terms.

Laplace Transform Table

A Laplace transform table is an invaluable tool for engineers and mathematicians working with differential equations and complex systems. It lists many common functions along with their Laplace transforms, enabling quick lookup and application of these transformations to various problems, including inverse Laplace transforms.

Using the Table for Inverse Transforms

To use a Laplace transform table for inverse transforms, first, ensure that the function you're working with matches a form in the table. For inverse transforms, you're essentially reversing the Laplacian process to find the time-domain function from a given s-domain function. In our textbook solution, after employing partial fraction decomposition, we turned to the table to identify the corresponding time-domain functions that matched the s-domain functions we obtained.

Comparing Coefficients

Comparing coefficients is a technique used in algebra to find values of unknown constants by matching the coefficients of corresponding powers of variables on both sides of an equation. It's based on the principle that if two polynomials are equal, then their corresponding coefficients for the same powers must also be equal.

Role in Solving for Constants

After decomposing a fraction, as we do with the partial fraction decomposition, we often end up with an equation with unknown constants. Multiplying out and consolidating terms gives us a polynomial equation. Then, by comparing coefficients of the same degree on both sides, we can find the values for those constants. The method requires careful attention to detail because a mistake in matching coefficients can lead to an incorrect solution. As seen in our exercise, this method was essential to determine the values of A and B, which allowed us to find the correct inverse Laplace transform.