27E In Problems, determine聽L-1F聽s2... [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: 27E (page 350)

In Problems, determine $L^{- 1} \{F\}$ $s^{2} F (s) - 4 F (s) = \frac{5}{s + 1}$

Short Answer

Expert verified

$L^{- 1} \{\frac{5}{(s^{2} - 4) (s + 1)}\} = \frac{5}{12} e^{2 t} + \frac{5}{4} e^{- 2 t} - \frac{5}{3} e^{- t}$

Step by step solution

Find the factor of the denominator

From the given equation we get:

$F (s) = \frac{5}{(s^{2} - 4) (s + 1)} = \frac{5}{(s - 2) (s + 2) (s + 1)}$

Using partial fractions, we get:

$\frac{5}{(s - 2) (s + 1) (s - 1)} = \frac{A}{s - 2} + \frac{B}{s + 2} + \frac{C}{s + 1}$

from where it follows:

$5 = A (s + 2) (s + 1) + B (s - 2) (s + 1) + C (s^{2} - 4)$

Using $s = 2, - 2, - 1,$ respectively we get:

$s = 2 : 5 = 12 A 鈬� A = \frac{5}{12} s = - 2 : 5 B = 4 鈬� B = \frac{5}{4} s = - 1 : 5 = - 3 C 鈬� C = - \frac{5}{3} T h e r e f o r e, \frac{5}{(s^{2} - 4) (s + 1)} = \frac{5}{12 (s - 2)} + \frac{5}{4 (s + 2)} - \frac{5}{3 (s + 1)}$

Find the inverse

Find the inverse Laplace transform we get:

$L^{- 1} \{\frac{5}{(s^{2} - 4) (s + 1)}\} = L^{- 1} \{\frac{5}{12 (s - 2)}\} + L^{- 1} \{\frac{5}{4 (s + 2)}\} + L^{- 1} \{\frac{5}{3 (s + 1)}\} = \frac{5}{12} L^{- 1} \{\frac{1}{s - 2}\} + \frac{5}{4} L^{- 1} \{\frac{1}{s + 2}\} - \frac{5}{3} L^{- 1} \{\frac{1}{s + 1}\} = \frac{5}{12} e^{2 t} + \frac{5}{4} e^{- 2 t} - \frac{5}{3} e^{- t} T h e r e f o r e, L^{- 1} \{\frac{5}{(s^{2} - 4) (s + 1)}\} = \frac{5}{12} e^{2 t} + \frac{5}{4} e^{- 2 t} - \frac{5}{3} e^{- t}$

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91影视

In Problems, determine $L^{- 1} \{F\}$ $s^{2} F (s) - 4 F (s) = \frac{5}{s + 1}$

Short Answer

Step by step solution

Find the factor of the denominator

Find the inverse

One App. One Place for Learning.

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91影视

In Problems, determineL-1Fs2Fs-4Fs=5s+1

Short Answer

Step by step solution

Find the factor of the denominator

Find the inverse

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Geometry

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems, determine $L^{- 1} \{F\}$ $s^{2} F (s) - 4 F (s) = \frac{5}{s + 1}$