35E Theorem 6 in Section 7.3 on page... [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: 35E (page 350)

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as
$L^{- 1} \{\frac{d^{n} F}{d s^{n}}\} (t) = {(- t)}^{n} f (t)$ ,
Whererole="math" localid="1664423939060" $f = L^{- 1} (F)$ .Use this equationin Problems 33-36 to compute $L^{- 1} (F) .$ $F (s) = l n (\frac{s^{2} + 9}{s^{2} + 1})$

Short Answer

Expert verified

$L^{- 1} (F) = \frac{2 c o s t - 2 c o s 3 t}{t}$

Step by step solution

Simplify the function and find derivative

Using the property of function $\ln t$ we get:

$F (s) = \ln (\frac{s^{2} + 9}{s^{2} + 9}) = \ln (s^{2} + 9) - \ln (s^{2} + 1)$

Find the derivative of F with respect to s:

$\frac{d F}{d s} = \frac{d}{d s} (\ln (s^{2} + 9) - \ln (s^{2} + 1)) \frac{d F}{d s} = \frac{d}{d s} (\ln (s^{2} + 9)) - \frac{d}{d s} (\ln (s^{2} + 1)) \frac{d F}{d s} = \frac{2 s}{s^{2} + 9} - \frac{2 s}{s^{2} + 1}$

Find the Laplace inverse

From the given condition, we have $L^{- 1} (F) = \frac{1}{{(- t)}^{n}} L^{- 1} \{\frac{d^{n} F}{d s^{n}}\}$ ,apply this to find the Laplace inverse as:

$L^{- 1} (F) = \frac{1}{(- t)} L^{- 1} \{\frac{d F}{d s}\} L^{- 1} (F) = \frac{1}{(- t)} L^{- 1} (\frac{2 s}{s^{2} + 9} - \frac{2 s}{s^{2} + 1}) L^{- 1} (F) = \frac{1}{t} (- L^{- 1} (\frac{2 s}{s^{2} + 9}) + L^{- 1} (\frac{2 s}{s^{2} + 1})) L^{- 1} (F) = \frac{1}{t} (- 2 \cos 3 t + 2 \cos t) L^{- 1} (F) = \frac{2 \cos t - 2 \cos 3 t}{t} T h e r e f o r e, L^{- 1} (F) = \frac{2 \cos t - 2 \cos 3 t}{t}$

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Short Answer

Step by step solution

Simplify the function and find derivative

Find the Laplace inverse

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91影视

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform asL-1dnFdsnt=-tnft,Whererole="math" localid="1664423939060" f=L-1F.Use this equationin Problems 33-36 to computeL-1F.Fs=lns2+9s2+1

Short Answer

Step by step solution

Simplify the function and find derivative

Find the Laplace inverse

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Geometry

Discrete Mathematics

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.