Use Laplace transform properties to determine
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To solve the Laplace transform of the given expression using Laplace transform properties, let's proceed step by step.

### Problem Statement

Determine the Laplace transform \( \mathcal{L} \left\{ t \frac{d}{dt} \left[ \cosh(2t) \right] \right\} \).

### Solution

Given expression: 
\[
\mathcal{L} \left\{ t \frac{d}{dt} \left[ \cosh(2t) \right] \right\}
\]

### Step 1: Identify the derivative property of the Laplace transform

The Laplace transform of the derivative of a function is given by:
\[
\mathcal{L} \left\{ \frac{d}{dt} f(t) \right\} = sF(s) - f(0)
\]
where \( F(s) \) is the Laplace transform of \( f(t) \), i.e., \( F(s) = \mathcal{L} \{ f(t) \} \).

### Step 2: Compute the derivative

First, we compute the derivative \( \frac{d}{dt} \left[ \cosh(2t) \right] \):

\[
\frac{d}{dt} \left[ \cosh(2t) \right] = 2\sinh(2t)
\]

Thus, the expression becomes:
\[
\mathcal{L} \left\{ t \cdot 2\sinh(2t) \right\}
\]

### Step 3: Use the time multiplication property of Laplace transform

The time multiplication property states:
\[
\mathcal{L} \left\{ t f(t) \right\} = -\frac{d}{ds} F(s)
\]
where \( F(s) = \mathcal{L} \{ f(t) \} \).

### Step 4: Find the Laplace transform of \( \sinh(2t) \)

The Laplace transform of \( \sinh(at) \) is:
\[
\mathcal{L} \left\{ \sinh(at) \right\} = \frac{a}{s^2 - a^2}
\]
For \( a = 2 \), we get:
\[
\mathcal{L} \left\{ \sinh(2t) \right\} = \frac{2}{s^2 - 4}
\]

### Step 5: Apply the time multiplication property

Using the time multiplication property:
\[
\mathcal{L} \left\{ t \cdot 2\sinh(2t) \right\} = -\frac{d}{ds} \left[ 2 \cdot \frac{2}{s^2 - 4} \right]
\]

### Step 6: Differentiate the function

Now, differentiate the function:
\[
-\frac{d}{ds} \left[ \frac{4}{s^2 - 4} \right] = -\frac{4 \cdot (-2s)}{(s^2 - 4)^2} = \frac{8s}{(s^2 - 4)^2}
\]

### Final Answer

Thus, the Laplace transform of the given expression is:
\[
\mathcal{L} \left\{ t \frac{d}{dt} \left[ \cosh(2t) \right] \right\} = \frac{8s}{(s^2 - 4)^2}
\]

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Would you like more details on any specific part, or do you have any questions?

### Related Questions
1. What is the Laplace transform of \( t \cdot \sinh(at) \)?
2. How can the differentiation property of the Laplace transform be applied to \( e^{at} \)?
3. What is the significance of the time multiplication property in the Laplace transform?
4. How do you compute the Laplace transform of a second derivative?
5. What is the Laplace transform of \( t^n \cdot \cosh(bt) \) for any integer \( n \)?

### Tip:
When dealing with time-domain multiplication by \( t \) in Laplace transforms, always remember to use the derivative property in the \( s \)-domain.

Learn how to determine the Laplace transform of t times the derivative of hyperbolic cosine function cosh(2t) using Laplace transform properties. This problem involves advanced concepts in Laplace transforms, derivatives, and hyperbolic functions. Suitable for advanced undergraduate students and professionals.

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