Chapter 7: Q7RP (page 415)

In Problems 3–10, determine the Laplace transform of the given function.
$7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t$

Short Answer

Expert verified

$L {7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t} (s) = \frac{7 (s - 2)}{{(s - 2)}^{2} + 9} - \frac{10}{{(s - 7)}^{2} + 25}$

Step by step solution

Step 1:Given Information

The given function is $7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t$

Determining the Laplace transform

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

$\begin{array}{l} L {e^{a t} s i n b t} (s) = \frac{b}{{(s - a)}^{2} + b^{2}} \\ L {e^{a t} c o s b t} (s) = \frac{s - a}{{(s - a)}^{2} + b^{2}} \end{array}$

Considering the Laplace transform's linearity, we get

$\begin{matrix} L {7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t} (s) = 7 L {e^{2 t} c o s 3 t} (s) - 2 L {e^{7 t} s i n 5 t} (s) \\ = 7 \frac{s - 2}{{(s - 2)}^{2} + 3^{2}} - 2 \frac{5}{{(s - 7)}^{2} + 5^{2}} \\ = \frac{7 (s - 2)}{{(s - 2)}^{2} + 9} - \frac{10}{{(s - 7)}^{2} + 25} \end{matrix}$

Determining the Result

Thus, the required Laplace transform is

$L {7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t} (s) = \frac{7 (s - 2)}{{(s - 2)}^{2} + 9} - \frac{10}{{(s - 7)}^{2} + 25}$

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In Problems 3–10, determine the Laplace transform of the given function.
$7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t$

Short Answer

Step by step solution

Step 1:Given Information

Determining the Laplace transform

Determining the Result

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In Problems 3–10, determine the Laplace transform of the given function.7e2tcos3t-2e7tsin5t

Short Answer

Step by step solution

Step 1:Given Information

Determining the Laplace transform

Determining the Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Probability and Statistics

Calculus

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Geometry

Study anywhere. Anytime. Across all devices.

In Problems 3–10, determine the Laplace transform of the given function.
$7 e^{2 t} c o s 3 t - 2 e^{7 t} s i n 5 t$