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:

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

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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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

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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

Calculus

Geometry

Probability and Statistics

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

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