Find the Laplace transform of ( s(t) = e^{-t} \sin(2t) \cosh(3t) using frequency shift

To find the Laplace transform of the given function \( s(t) = e^{-t} \sin(2t) \cosh(3t) \), we will use the frequency shift property of the Laplace transform.

The **frequency shift theorem** states:
\[
\mathcal{L}\{e^{at} f(t)\} = F(s - a)
\]
where \( F(s) \) is the Laplace transform of \( f(t) \), and \( a \) is a constant.

### Step 1: Simplify \( s(t) \)
We are given:
\[
s(t) = e^{-t} \sin(2t) \cosh(3t)
\]

First, let's express \( \cosh(3t) \) in terms of exponentials:
\[
\cosh(3t) = \frac{e^{3t} + e^{-3t}}{2}
\]

Substitute this into the original function:
\[
s(t) = e^{-t} \sin(2t) \cdot \frac{e^{3t} + e^{-3t}}{2}
\]
\[
s(t) = \frac{1}{2} e^{-t} \left( \sin(2t) e^{3t} + \sin(2t) e^{-3t} \right)
\]
\[
s(t) = \frac{1}{2} \left( e^{2t} \sin(2t) + e^{-4t} \sin(2t) \right)
\]

### Step 2: Apply the frequency shift property
We need to find the Laplace transform of each term separately.

1. **For \( e^{2t} \sin(2t) \)**:
   Using the frequency shift theorem and the standard Laplace transform of \( \sin(2t) \), we have:
   \[
   \mathcal{L}\{\sin(2t)\} = \frac{2}{s^2 + 4}
   \]
   Therefore, applying the frequency shift \( e^{2t} \), we shift the argument by \(-2\):
   \[
   \mathcal{L}\{e^{2t} \sin(2t)\} = \frac{2}{(s - 2)^2 + 4}
   \]

2. **For \( e^{-4t} \sin(2t) \)**:
   Similarly, applying the frequency shift theorem for \( e^{-4t} \), we shift the argument by \( 4 \):
   \[
   \mathcal{L}\{e^{-4t} \sin(2t)\} = \frac{2}{(s + 4)^2 + 4}
   \]

### Step 3: Combine the results
Now, we combine the Laplace transforms of both terms:
\[
\mathcal{L}\{s(t)\} = \frac{1}{2} \left( \frac{2}{(s - 2)^2 + 4} + \frac{2}{(s + 4)^2 + 4} \right)
\]
\[
\mathcal{L}\{s(t)\} = \frac{1}{(s - 2)^2 + 4} + \frac{1}{(s + 4)^2 + 4}
\]

Thus, the Laplace transform of \( s(t) = e^{-t} \sin(2t) \cosh(3t) \) is:
\[
\boxed{\mathcal{L}\{s(t)\} = \frac{1}{(s - 2)^2 + 4} + \frac{1}{(s + 4)^2 + 4}}
\]

Would you like further details on any of the steps?

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Here are 5 related questions to deepen your understanding:

1. How do you derive the Laplace transform of \( \sin(at) \)?
2. What is the Laplace transform of \( \cosh(bt) \)?
3. How can the frequency shift theorem be applied to more complex functions?
4. What is the inverse Laplace transform of \( \frac{1}{(s - a)^2 + b^2} \)?
5. How does the Laplace transform help solve differential equations?

**Tip**: The frequency shift property is very powerful when handling exponentials in front of simpler functions like sines or cosines.

Find the Laplace transform of s(t) = e^{-t} \sin(2t) \cosh(3t) using frequency shift

Explore how to find the Laplace transform of s(t) = e^{-t} \sin(2t) \cosh(3t) using the frequency shift theorem. This detailed step-by-step solution involves applying the frequency shift property, handling hyperbolic and trigonometric functions. Suitable for undergraduate students studying applied mathematics.

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