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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: 33E (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 $L^{- 1} \{F\} .$ $F (s) = l n (\frac{s + 2}{s - 5})$

Short Answer

Expert verified

$L^{- 1} (F) = \frac{1}{t} (e^{5 t} - e^{2 t})$

Step by step solution

01

Simplify the function and find derivative

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

$F (s) = \ln (\frac{s + 2}{s - 5}) F (s) = \ln (s + 2) - \ln (s - 5)$

Find the derivative of F with respect to s:

$\frac{d F}{d s} = \frac{d}{d s} (\ln (s + 2)) - \frac{d}{d s} (\ln (s - 5)) \frac{d F}{d s} = \frac{1}{s + 2} - \frac{1}{s - 5}$

02

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:

$L^{- 1} (F) = \frac{1}{{(- t)}^{1}} L^{- 1} \{\frac{d F}{d s}\} L^{- 1} (F) = \frac{1}{(- t)} L^{- 1} (\frac{1}{s + 2} - \frac{1}{s - 5}) = \frac{1}{t} (- L^{- 1} (\frac{1}{s + 2}) + L^{- 1} (\frac{1}{s - 5})) = \frac{1}{t} (e^{5 t} - e^{2 t}) T h e r e f o r e, L^{- 1} (F) = \frac{1}{t} (e^{5 t} - e^{2 t})$

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Most popular questions from this chapter

Question: In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.

$y'' + 6 y = t^{2} - 1; y (0) = 0, y' (0) = - 1$

Determine the inverse Laplace transform of the given function.

$\frac{1}{s^{2} + 4 s + 8}$ .

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.]

$e^{- 2 t} \sin 2 t + e^{3 t} t^{2}$

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.]

$\sin^{2} t$ .

The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y(t) to the Laplace transform of the input function g(t), when all initial conditions are zero. If a linear system is governed by the differential equation

$y'' (t) + 6 y' (t) + 10 y (t) = g (t), t > 0$

use the linearity property of the Laplace transform and Theorem 5 on page363 on the Laplace transform of higher-order derivatives to determine the transfer function $H (s) = Y (s) / G (s)$ of this system.

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