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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 415

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

In Problems 3–10, determine the Laplace transform of the given function.
$e^{2 t} - t^{3} + t^{2} - s i n 5 t$

$L {e^{2 t} - t^{3} + t^{2} - s i n 5 t} = \frac{1}{s - 2} - \frac{6}{s^{4}} + \frac{2}{s^{3}} - \frac{5}{s^{2} + 25}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Given Information

The given function is . $e^{2 t} - t^{3} + t^{2} - s i n 5 t$

Step 2: Determining the Laplace transform:

Using the following Laplace transform definition, to find Laplace of given function.

$\begin{matrix} L {e^{a t}} (s) = \frac{1}{s - a} \\ L {t^{a}} (s) = \frac{a!}{s^{a + 1}} \\ L {s i n a t} (s) = \frac{a}{s^{2} + a^{2}} \end{matrix}$

Considering the Laplace transform's linearity, we get

$\begin{matrix} L {e^{2 t} - t^{3} + t^{2} - s i n 5 t} (s) = L {e^{2 t}} (s) - L {t^{3}} (s) + L {t^{2}} (s) - L {s i n 5 t} (s) \\ = \frac{1}{s - 2} - \frac{3!}{s^{4}} + \frac{2!}{s^{3}} - \frac{5}{s^{2} + 5^{2}} \\ = \frac{1}{s - 2} - \frac{0}{s^{4}} + \frac{z}{s^{3}} - \frac{5}{s^{2} + 25} \end{matrix}$

Step 3: Determining the Result

Thus, the required Laplace transform is . $L {e^{2 t} - t^{3} + t^{2} - s i n 5 t} = \frac{1}{s - 2} - \frac{6}{s^{4}} + \frac{2}{s^{3}} - \frac{5}{s^{2} + 25}$

Most popular questions for Math Textbooks

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

$10 . y'' - 4 y = 4 t - 8 e^{- 2 t}; y (0) = 0, y' (0) = 5$

In problem 15-22, solve the given integral equation or integro differential equation for $y (t)$

$L \int_{0}^{t} (t - v) y (v) d v = 1$

In Problems 11–20, determine the partial fraction expansion for the given rational function.

$\frac{- 8 s^{2} - 5 s + 9}{(s + 1) (s^{2} - 3 s + 2)}$

In Problems 29 - 32, use the method of Laplace transforms to find a general solution to the given differential equation by assuming $y (0) = a a n d y' (0) = b$ a and b are arbitrary constants.

$y'' - 5 y' + 6 y = - 6 t e^{2 t}$

Find the Laplace transform of $f (t) = \int_{0}^{t} (t - v) e^{3 v} d v$

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