Q2RP In Problems 1 and 2, use the def... [FREE SOLUTION]

91影视

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: Q2RP (page 415)

In Problems 1 and 2, use the definition of the Laplace transform to determine $L {f}$ .
$f (t) = {\begin{cases} e^{- t}, 0 鈮� t 鈮� 5 \\ - 1, 5 < t \end{cases}$

Short Answer

Expert verified

$L {f (t)} (s) = \frac{1}{s + 1} (1 - e^{- 5 (s + 1)}) - \frac{1}{s} e^{- 5 s}$

Step by step solution

Step 1:Given Information

The given function is $f (t) = {\begin{cases} e^{- t}, 0 鈮� t 鈮� 5 \\ - 1, 5 < t \end{cases}$ .

Determining the

Using the Laplace transform definition, we get

$\begin{matrix} L {f (t)} (s) = {鈭�}_{0}^{鈭�} e^{- s t} f (t) d t \\ = {鈭�}_{0}^{5} e^{鈭� s t} e^{鈭� t} d t + {鈭�}_{5}^{鈭�} (鈭� 1) e^{鈭� s t} d t \\ = {鈭�}_{0}^{5} e^{鈭� t (s + 1)} d t 鈭� \lim_{N 鈫� 鈭�} {鈭�}_{5}^{N} e^{鈭� s t} d t \end{matrix}$

Simplify further as:

$\begin{matrix} L {f (t)} = {[鈭� \frac{e^{鈭� t (s + 1)}}{s + 1}]}_{0}^{5} 鈭� \lim_{N 鈫� 鈭�} {[鈭� \frac{e^{鈭� s t}}{s}]}_{5}^{N} \\ = \frac{1}{s + 1} (1 鈭� e^{鈭� 5 (s + 1)}) 鈭� \lim_{N 鈫� 鈭�} (\frac{1}{s} (e^{鈭� 5 s} 鈭� e^{鈭� N s})) \\ = \frac{1}{s + 1} (1 鈭� e^{鈭� 5 (s + 1)}) 鈭� \frac{1}{s} e^{鈭� 5 s} \end{matrix}$

Determining the Result

Thus, the required Laplace transform is $L {f (t)} (s) = \frac{1}{s + 1} (1 - e^{- 5 (s + 1)}) - \frac{1}{s} e^{- 5 s}$

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91影视

In Problems 1 and 2, use the definition of the Laplace transform to determine $L {f}$ .
$f (t) = {\begin{cases} e^{- t}, 0 鈮� t 鈮� 5 \\ - 1, 5 < t \end{cases}$

Short Answer

Step by step solution

Step 1:Given Information

Determining the

Determining the Result

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91影视

In Problems 1 and 2, use the definition of the Laplace transform to determine L{f}.f(t)={e-t,0鈮�t鈮�5-1,5<t

Short Answer

Step by step solution

Step 1:Given Information

Determining the

Determining the Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Mechanics Maths

Statistics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 1 and 2, use the definition of the Laplace transform to determine $L {f}$ .
$f (t) = {\begin{cases} e^{- t}, 0 鈮� t 鈮� 5 \\ - 1, 5 < t \end{cases}$