Q7RP

Question

In Problems 3–10, determine the Laplace transform of the given function.

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

Step-by-Step Solution

Verified

Answer

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

1Step 1:Given Information

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

2Step 2: 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

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

3Step 3: 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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