Question

Find the inverse transform of each of the following functions: (a) \(\frac{s}{\left(s^{2}+1\right)\left(s^{2}+2\right)}\). (d) \(\frac{1}{s(s+1)^{2}}\). (b) \(\frac{1}{\left(s^{2}+1\right)\left(s^{2}+2\right)}\). (e) \(\frac{1}{s(s+1)}\). (c) \(\frac{1}{(s-1)(s-2)}\). (f) \(\frac{s-4}{\left(s^{2}+4\right)^{2}}\).

Short answer

Question: Find the inverse transform of the function \(\frac{s}{\left(s^{2}+1\right)\left(s^{2}+2\right)}\) Solution: The inverse Laplace transform of the function is given by: \(f(t) = \mathcal{L}^{-1}\left\{\frac{As + B}{s^{2}+1}\right\} + \mathcal{L}^{-1}\left\{\frac{Cs + D}{s^{2}+2}\right\}\) To find the coefficients A, B, C, and D, use partial fraction decomposition, differentiate, and plug in specific values of s. After finding A, B, C, and D, plug them back into the equation and perform the inverse Laplace transforms. This will give you the inverse Laplace transform of the function in terms of time (t).

Step-by-step solution

Perform Partial Fraction Decomposition

Write the given function, \(\frac{s}{\left(s^{2}+1\right)\left(s^{2}+2\right)}\), in the form of partial fractions: $$\frac{s}{\left(s^{2}+1\right)\left(s^{2}+2\right)} = \frac{As + B}{s^{2}+1} + \frac{Cs + D}{s^{2}+2}$$

Clear the Denominator and Combine

Multiply both sides by the common denominator \(\left(s^{2}+1\right)\left(s^{2}+2\right)\) to clear the denominators, then combine terms: $$s = (As + B)(s^{2}+2) + (Cs + D)(s^{2}+1)$$

Find the Coefficients

To find the coefficients A, B, C, and D, we will substitute specific values of s that will simplify the equation and help us find each coefficient one by one. For A: Choose s = 0, then the equation becomes $$0 = B(2) + D(1)$$ For B: Differentiate both sides of the main equation with respect to s and then plug in s = 0. For C and D: Plug in s = ±i or s = ±\sqrt{2}i into the equation and solve using either substitution or elimination method.

Inverse Laplace Transform

After finding the coefficients A, B, C, and D, plug them back into the partial fraction decomposition equation and perform the inverse Laplace transform of each term using well-known inverse Laplace transforms and properties. Let f(t) be the inverse Laplace transform of the given function: $$f(t) = \mathcal{L}^{-1}\left\{\frac{As + B}{s^{2}+1}\right\} + \mathcal{L}^{-1}\left\{\frac{Cs + D}{s^{2}+2}\right\}$$ By following the steps above, you will obtain the inverse Laplace transform for part (a). To solve the remaining parts (b)-(f), repeat the same steps using the corresponding functions provided.

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify complex rational expressions, particularly when working with the inverse Laplace transform. It involves breaking down a fraction into a sum of simpler fractions, with the goal of making the inverse transformation easier to manage.

The first step is to express the given fraction as a sum of fractions with simpler denominators. For instance, a rational expression like \( \frac{s}{(s^{2}+1)(s^{2}+2)} \) is transformed into a sum of fractions where the denominators are the factors of the original denominator, and the numerators are yet to be determined constants or linear polynomials. The structure of these simpler fractions is based on the nature of the factors in the original denominator.

To find the unknowns in the numerators, which are generally denoted as A, B, C, D in the general form, various techniques can be employed such as substituting specific values for 's' or differentiating both sides of the equation. Once these coefficients are determined, each term in the decomposition may correspond to a known inverse Laplace transform, which greatly simplifies the calculation of the transformation back into the time domain.

Laplace Transform Properties

One of the reasons why the Laplace transform is so powerful in solving differential equations is due to its properties that ease the transformation of complex functions. Some of these properties include linearity, the first and second shifting theorems, and the property for inverse transforms.

The linearity property states that the Laplace transform of a sum is equal to the sum of the Laplace transforms, and it is also true for the inverse: \( \mathcal{L}^{-1}\{cF(s) + dG(s)\} = c\mathcal{L}^{-1}\{F(s)\} + d\mathcal{L}^{-1}\{G(s)\} \), where c and d are constants. This becomes particularly useful after applying partial fraction decomposition.

The shifting theorems adjust the function for exponential factors, and the inverse Laplace transform property ensures that transforming back to the time domain is feasible. Understanding and applying these properties appropriately can significantly streamline solving problems involving the Laplace transform.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In practical terms, they are used to describe the behavior of complex systems, such as electrical circuits, mechanical systems, or even natural phenomena.

The connection between differential equations and the Laplace transform lies in the Laplace transform's ability to convert differential equations into algebraic equations, which are much simpler to solve. Once an algebraic solution is found, the inverse Laplace transform is used to revert back to the original domain providing the solution to the differential equation.

For students working with differential equations, understanding how to navigate from the time domain to the s-domain through the Laplace transform and back again using the inverse Laplace transform is critical. Mastery of partial fraction decomposition and the properties of the Laplace transform makes this process more approachable and less daunting, underscoring the importance of these mathematical tools in solving differential equations.