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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. $t^{2} e^{- 9 t}$

$L {t^{2} e^{- 9 t}} (s) = \frac{2}{{(s + 9)}^{3}}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Given Information.

The given function is . $t^{2} e^{- 9 t}$

Step 2: Determining the Laplace transform:

Using the Laplace transform definition, $L {t^{n} e^{a t}} = \frac{n!}{{(s - a)}^{n + 1}}$ we get

$\begin{matrix} L {t^{2} e^{- 9 t}} = \frac{2!}{{(s + 9)}^{3}} \\ = \frac{2}{{(s + 9)}^{3}} \end{matrix}$

Step 3: Determining the Result

Thus, the required Laplace transform is . $L {t^{2} e^{- 9 t}} (s) = \frac{2}{{(s + 9)}^{3}}$

Most popular questions for Math Textbooks

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.]

$(t - 1)^{4}$ .

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 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.]

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

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 1-20, 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.]

$3 t^{2} - e^{2 t}$

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