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Q14E

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 404

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

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

The Laplace transformation of $\int_{0}^{t} e^{v} \sin (t - v) d v$ is $\frac{1}{(s^{2} + 1) (s - 1)}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given that,

$f (t) = \int_{0}^{t} e^{v} \sin (t - v) d v$

Step 2: Applying the Convolution Theorem

$g ⋆ h = \int_{0}^{t} g (t - v) h (v) d v g (t) = \sin t g (s) = \frac{1}{s^{2} + 1} h (t) = e^{t} g (s) = \frac{1}{s - 1}$

Step 3: Simplify

$\begin{matrix} L [g ⋆ h] = g (s) \cdot h (s) \\ = \frac{1}{s^{2} + 1} \cdot \frac{1}{s - 1} \\ = \frac{1}{(s^{2} + 1) (s - 1)} \end{matrix}$

Hence,

$f (s) = \frac{1}{(s^{2} + 1) (s - 1)}$

Most popular questions for Math Textbooks

Determine the inverse Laplace transform of the given function.

$\frac{6}{{(s - 1)}^{4}}$ .

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

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

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

$y'' - 7 y' + 10 y = 9 cost + 7 sint; y (0) = 5, y' (0) = - 4$

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.]

$cosnt sinmt, m \neq n$

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.]

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

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