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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 416

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Determine the inverse Laplace transform of the given function.
$\frac{2 s - 1}{s^{2} - 4 s + 6}$

The solution is

$L^{- 1} \{\frac{2 (s - 2) + 3}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}}\} (t) = 2 e^{2 t} \cos \sqrt{2} t + \frac{3}{\sqrt{2}} e^{2 t} \sin \sqrt{2} t$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

The given function value in s domain is $\frac{2 s - 1}{s^{2} - 4 s + 6}$

Step 2: Use partial fractions

Factorize the denominator of the given function as

$\begin{matrix} s^{2} - 4 s + 6 = s^{2} - 4 s + 4 + 2 \\ = {(s - 1)}^{2} + {(\sqrt{2})}^{2} \end{matrix}$

The function becomes. $\frac{2 s - 1}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}}$ Decompose the function as:

$\begin{matrix} \frac{2 s - 1}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}} = \frac{2 (s - 2) + 3}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}} \\ = \frac{2 (s - 2)}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}} + \frac{3}{\sqrt{2}} \frac{\sqrt{2}}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}} \end{matrix}$

Take inverse Laplace transform using $L^{- 1} \{\frac{s - a}{{(s - a)}^{2} + b^{2}}\} (t) = e^{a t} \cos b t$ and $L^{- 1} \{\frac{b}{{(s - a)}^{2} + b^{2}}\} (t) = e^{a t} \sin b t$ as:

$L^{- 1} \{\frac{2 (s - 2) + 3}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}}\} (t) = 2 e^{2 t} \cos \sqrt{2} t + \frac{3}{\sqrt{2}} e^{2 t} \sin \sqrt{2} t$

Therefore,the required inverse Laplace transform is:

$L^{- 1} \{\frac{2 (s - 2) + 3}{{(s - 2)}^{2} + {(\sqrt{2})}^{2}}\} (t) = 2 e^{2 t} \cos \sqrt{2} t + \frac{3}{\sqrt{2}} e^{2 t} \sin \sqrt{2} t$

Most popular questions for Math Textbooks

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

$y'' - 3 y' + 2 y = cost; y (0) = 0, y' (0) = - 1$

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.]

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

In Problems 1-10, determine the inverse Laplace transform of the given function.

$\frac{s - 1}{2 s^{2} + s + 6}$

Find the Laplace transform of . $f (t) = \int_{0}^{t} e^{v} s i n (t - v) d v$

Use the method of Laplace transforms to solve

$\begin{array}{l} x^{''} + y^{'} = 2; & x (0) = 3 \begin{matrix} , & x^{'} (0) = 0 \end{matrix} \\ 4 x + y^{'} = 6; & y (1) = 4 . \end{array}$

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