Q7.3-3E In Problems 1-20, determine the ... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 7: Q7.3-3E (page 365)

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]
$e^{- t} \cos 3 t + e^{6 t} - 1$ .

Short Answer

Expert verified

The Laplace transform for the given equation is $\frac{s + 1}{{(s + 1)}^{2} + 9} + \frac{1}{s - 6} - \frac{1}{s} for s > 6$ .

Step by step solution

Definition of Laplace transform

The integral transform of a given derivative function with real variable t into a complex function with variable s is known as the Laplace transform.
Let f(t) be supplied for t(0), and assume that the function meets certain constraints that will be presented subsequently.
The Laplace transform formula defines the Laplace transform of f(t), which is indicated by $L \{f (t)\}$ or F(s).

Determine the Laplace transform for the given equation

Given that $e^{- t} \cos 3 t + e^{6 t} - 1$ .

Find the Laplace transform of the given function $e^{- t} \cos 3 t + e^{6 t} - 1$ using the Laplace formula

$L \{a f_{1} + b f_{2}\} = a L \{f_{1}\} + b L \{f_{2}\}$ , $L \{e^{a t} c o s b t\} = \frac{s - a}{{(s - a)}^{2} + b^{2}}$ , $L {1} = \frac{1}{s}$ , $L \{t^{n}\} = \frac{n!}{s^{n + 1}}$ and $L \{e^{a t}\} = \frac{1}{s - a}$ as follows:

$\begin{array}{rcl} L \{e^{- t} \cos 3 t + e^{6 t} - 1\} & = & L \{e^{- t} \cos 3 t\} + L \{e^{6 t}\} - L {1} \\ = & \frac{s - (- 1)}{{(s - (- 1))}^{2} + 3^{2}} + \frac{1}{s - 6} - \frac{1}{s} \\ = & \frac{s + 1}{{(s + 1)}^{2} + 9} + \frac{1}{s - 6} - \frac{1}{s} for s > 6 \end{array}$

Therefore, the Laplace transform for the given equation is

$\frac{s + 1}{{(s + 1)}^{2} + 9} + \frac{1}{s - 6} - \frac{1}{s} for s > 6$

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91影视

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]
$e^{- t} \cos 3 t + e^{6 t} - 1$ .

Short Answer

Step by step solution

Definition of Laplace transform

Determine the Laplace transform for the given equation

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91影视

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]e-tcos3t+e6t-1.

Short Answer

Step by step solution

Definition of Laplace transform

Determine the Laplace transform for the given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Discrete Mathematics

Pure Maths

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]
$e^{- t} \cos 3 t + e^{6 t} - 1$ .