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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: 34E (page 350)

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as
$L^{- 1} \{\frac{d^{n} F}{d s^{n}}\} (t) = {(- t)}^{n} f (t),$
Where $f = L^{- 1} \{F\}$ .Use this equation in Problems 33-36 to compute $L^{- 1} \{F\}$ . $F (s) = l n (\frac{s - 4}{s - 3})$ .

Short Answer

Expert verified

$L^{- 1} (F) = \frac{1}{t} (e^{3 t} - e^{4 t})$

Step by step solution

01

Simplify the function and find derivative

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

$F (s) = \ln (\frac{s - 4}{s - 3}) F (s) = \ln (s - 4) - \ln (s - 3)$

Find the derivative of F with respect to s:

$\frac{d F}{d s} = \frac{d}{d s} (\ln (s - 4) - \ln (s - 3)) \frac{d F}{d s} = \frac{d}{d s} (\ln (s - 4)) - \frac{d}{d s} (\ln (s - 3)) \frac{d F}{d s} = \frac{1}{s - 4} - \frac{1}{s - 3}$

02

Find the Laplace inverse

From the given condition, we have $L^{- 1} (F) = \frac{1}{{(- t)}^{1}} L^{- 1} \{\frac{d F}{d s}\} L^{- 1} (F) = \frac{1}{(- t)} L^{- 1} (\frac{1}{s - 4} - \frac{1}{s - 3}) L^{- 1} (F) = \frac{1}{(t)} (- L^{- 1} (\frac{1}{s - 4}) + (\frac{1}{s - 3})) L^{- 1} (F) = \frac{1}{(t)} (e^{3 t^{}} - e^{4 t}) T h e r e f o r e, L^{- 1} (F) = \frac{1}{(t)} (e^{3 t^{}} - e^{4 t})$

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Most popular questions from this chapter

Recompute the coupled mass鈥搒pring oscillator motion in Problem 1, Exercises 5.6 (page 287), using Laplace transforms.

In Problems $1 - 14$ , solve the given initial value problem using the method of Laplace transforms.

$y'' + 6 y' + 5 y = 12 e^{t}; y (0) = - 1, y' (0) = 7$

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.]

$3 t^{4} - 2 t^{2} + 1$

In Problems 1鈥�19, use the method of Laplace transforms to solve the given initial value problem. Here x鈥�, y鈥�, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{l} D [x] + y = 0; 鈥夆赌夆赌� x (0) = 7 / 4, \\ 4 x + D [y] = 3; 鈥夆赌夆赌� y (0) = 4 \end{array}$

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.]

$2 t^{2} e^{- t} - t + \cos 4 t$ .

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