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Chapter 7: Problem 18

Find the given inverse transform. \(\mathscr{L}^{-1}\left\\{\frac{s-1}{s^{2}+2}\right\\}\)

Short Answer

Expert verified
The inverse Laplace transform is \( e^{t} \cos(\sqrt{2}t) \).

Step by step solution

01

Identify the Inverse Laplace Formula

The expression given is \( \frac{s-1}{s^2+2} \) and we need to find its inverse Laplace transform \( \mathscr{L}^{-1} \). Recognize that it is similar to the form \( \frac{s-a}{s^2+b^2} \), which is a translation of a damped cosine in the time domain.
02

Express the Function for Known Transform

Recall the Laplace inverse formulas: \( \mathscr{L}^{-1} \left\{ \frac{s-a}{s^2 + b^2} \right\} = e^{at} \cos(bt) \). Here, \( a = 1 \) and \( b = \sqrt{2} \).
03

Apply the Inverse Laplace Formula

Using the identified parameters \( a = 1 \) and \( b = \sqrt{2} \), the inverse would be \( e^{t} \cos(\sqrt{2}t) \). This is because the \( s-1 \) translates to \( e^{t} \) and the denominator \( s^2 + 2 \) equates the \( \cos(\sqrt{2}t) \) part.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laplace Transform
The Laplace Transform is a powerful mathematical tool used to solve differential equations by transforming functions from the time domain into the complex frequency domain. This transformation makes it easier to work with differential equations, as derivatives become algebraic expressions of complex frequency.
  • It is defined as: \( extbf{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) \, dt \)
  • By converting time-based functions into simpler algebraic forms, the Laplace Transform assists in analyzing and solving linear systems.
To understand it better, imagine converting a complex shape into its simpler geometric expression. Similarly, Laplace Transforms simplify the analysis of systems characterized by differential equations, promoting ease of manipulation before converting back with the inverse transform.
Damped Cosine
A Damped Cosine function describes a cosine wave that decreases in amplitude over time. It is commonly encountered in systems exhibiting oscillatory behavior with decreasing energy, such as a swinging pendulum coming to a stop.
  • Mathematically, a damped cosine function in the time domain is often expressed as \( e^{at} \cos(bt) \), where \( a \) represents the damping factor.
  • The damping factor, \( e^{at} \), causes the oscillations of the cosine to decrease exponentially.
The inverse Laplace transform \( \mathscr{L}^{-1} \left\{ \frac{s-a}{s^2+b^2} \right\} = e^{at} \cos(bt) \) illustrates how the frequency and damping interact. Recognizing such forms in transforms helps translate complex algebraic forms back into understandable time-domain functions.
Complex Frequency
Complex frequency, denoted by \( s \) in the Laplace domain, encompasses both a real and an imaginary component, represented as \( s = \sigma + j\omega \). These components provide insight into both exponential decay and oscillatory behavior of a system.
  • The real part \( \sigma \) correlates to growth or decay in the system, influencing whether the system's function rises or falls over time.
  • The imaginary part \( j\omega \) is related to the oscillation frequency, dictating how fast the system oscillates.
Understanding complex frequency is essential when interpreting transforms, as it affects the behavior of the inverse transform. It provides a bridge between the algebraic solutions in the frequency domain and their manifestation in the physical system, guiding where the system might oscillate or stabilize over time.

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Most popular questions from this chapter

Fill in the blanks or answer true/false. If \(f\) is not piecewise continuous on \([0, \infty)\), then \(\mathscr{L}\\{f(t)\\}\) will not exist._____

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{s^{2}+2 s+5}\right\\} $$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)=\cos t+\int_{0}^{t} e^{-t} f(t-\tau) d \tau $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{t} \cos ^{2} 3 t\right\\} $$

Fill in the blanks or answer true/false. \(1 \cdot 1=\)_____
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