Question

Find the inverse Laplace transform. $$F(s)=\frac{3 s^{2}+s+8}{s^{3}+4 s}$$

Short answer

Answer: The inverse Laplace transform of the given function F(s) is $$f(t) = 2+\sin(2t)+\frac{1}{2}\cos(2t)$$

Step-by-step solution

Perform partial fraction decomposition

In order to rewrite F(s) in terms of known Laplace transforms, we need to perform partial fraction decomposition. Given, $$F(s)=\frac{3 s^{2}+s+8}{s^{3}+4 s}$$ Let's rewrite F(s) as: $$F(s)=\frac{3 s^{2}+s+8}{s(s^2+4)}$$ Denoting the decomposition terms as A, B, and C: $$F(s) = \frac{A}{s}+\frac{Bs+C}{s^2+4}$$ Now, we will find the values of A, B, and C.

Find A, B, and C

To determine the values of A, B, and C, we follow these steps: 1. Multiply both sides of the equation by s(s^2+4) to get rid of denominators. $$3s^2+s+8=A(s^2+4)+(Bs+C)s$$ 2. Expand and rearrange the equation. $$3s^2+s+8=(A+B)s^2+Cs+4A$$ 3. Equate the coefficients of the powers of s. For s^2: 3 = A + B For s^1: 1 = C For s^0: 8 = 4A 4. Solve these equations to find A, B, and C. We can find A by solving (8=4A): $$A=2$$ We can find B by solving (3=A+B with A=2): $$B=1$$ Finally, we already have C: $$C=1$$ Now that we have A, B, and C, we can rewrite F(s) using these values.

Rewrite F(s) using A, B, and C

Using the values of A, B, and C, we have: $$F(s) = \frac{2}{s}+\frac{s+1}{s^2+4}$$

Apply the inverse Laplace transform

Now, we can finally find the inverse Laplace transform. We know that, $$f(t) = \mathcal{L}^{-1}\{F(s)\}$$ We apply the inverse Laplace transform to the simplified F(s) expression: $$f(t)=\mathcal{L}^{-1}\{\frac{2}{s}\}+\mathcal{L}^{-1}\{\frac{s+1}{s^2+4}\}$$ We know the inverse Laplace transforms of each term in F(s): $$f(t) = 2\mathcal{L}^{-1}\{\frac{1}{s}\}+\mathcal{L}^{-1}\{\frac{s}{s^2+4}\}+\mathcal{L}^{-1}\{\frac{1}{s^2+4}\}$$ $$f(t) = 2(1)+\sin(2t)+\frac{1}{2}\cos(2t)$$ $$f(t) = 2+\sin(2t)+\frac{1}{2}\cos(2t)$$ The inverse Laplace transform of the given function F(s) is: $$f(t) = 2+\sin(2t)+\frac{1}{2}\cos(2t)$$

Key concepts

Partial Fraction Decomposition

When dealing with complex rational expressions, especially within the context of the Laplace Transform, it's often necessary to break them down into simpler parts that can be managed more easily. This process is known as Partial Fraction Decomposition. It's a technique used to express the ratio of two polynomials as the sum of simpler rational expressions, typically with linear or quadratic denominators that are easier to invert if we apply an inverse Laplace Transform.

The general goal is to rewrite a complicated fraction as a sum of simpler terms, where each term has a polynomial numerator of lower degree than its corresponding denominator. To perform this decomposition, you equate the original expression to an unknown sum of partial fractions and then solve for the coefficients of these fractions. This often involves equating the coefficients of corresponding powers of the variable and solving a system of linear equations. Through this process, the inverse Laplace Transform becomes wieldier, transforming an intimidating algebraic expression into a combination of elemental functions.

Laplace Transform Properties

The Laplace Transform is an integral transform used extensively to solve differential equations by converting functions of time into functions of a complex variable. Understanding its properties can greatly simplify the process of finding inverse transforms. Several key properties are often used:

• Linearity: The Laplace Transform of a sum is the sum of the Laplace Transforms of the individual functions.
• Frequency shift: Shifting a function by a factor in the frequency domain corresponds to multiplying by an exponential in the time domain and vice versa.
• First Derivative: The Laplace Transform converts differentiation into a multiplication, which is particularly useful in solving linear ordinary differential equations.

Considering these properties, you will find it easier to handle the transformation of more complex functions and their eventual inversion. Each property provides a new tool in the mathematician’s toolkit to untangle the web of potentially cumbersome mathematical relations.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In engineering, physics, economics, and other sciences, the behaviors of systems are very often modeled by differential equations. These equations describe relationships between changing quantities and their rates of change.

The solutions to differential equations can reveal important information about the systems they model and often require specific techniques to solve. One powerful method is the use of Laplace Transforms, which transform differential equations into algebraic ones. The inverse Laplace Transform is then used to turn the algebraic solution back into a function of time. As demonstrated in the original exercise, the inverse Laplace Transform was applied after decomposing the functions using partial fractions, showing how multiple mathematical concepts interconnect to lead to a solution of a differential equation. Understanding each of these components is crucial in problem-solving within calculus and related fields.