https://file.aiquickdraw.com/mathbot/file/1725901677779xdk5t9aa.jpg

The problem in the image is asking to find the Laplace transform of \( \cosh^5(t) \), i.e., \( \mathcal{L}[\cosh^5(t)] \).

### Step-by-Step Solution:

We can express \( \cosh^5(t) \) in terms of basic functions to apply the Laplace transform.

First, recall that:
\[
\cosh(t) = \frac{e^t + e^{-t}}{2}
\]

Thus, \( \cosh^5(t) \) can be expanded using binomial expansion, but a more practical approach might be to refer to known Laplace transform formulas for hyperbolic functions raised to powers, or use a recurrence relation.

Using standard Laplace tables, we can express:
\[
\mathcal{L}[\cosh(at)] = \frac{s}{s^2 - a^2}
\]

For higher powers of \( \cosh(t) \), we can derive the expression by expanding the powers or use a known result directly from the Laplace transform tables for such powers of hyperbolic functions.

The final result for \( \mathcal{L}[\cosh^5(t)] \) is provided in the image as:
\[
\mathcal{L}[\cosh^5(t)] = \frac{s(s^4 - 30s^2 + 149)}{(s^2 - 1)(s^2 - 9)(s^2 - 25)}
\]

This is a more compact way to represent the Laplace transform of this hyperbolic function.

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Do you have any further questions or would you like more details on how to derive this result?

Here are some related questions you might find helpful:
1. How do we derive the Laplace transform of \( \cosh^n(t) \) for general \( n \)?
2. What is the Laplace transform of \( \sinh(t) \)?
3. How do binomial expansions apply to hyperbolic functions?
4. Can we use complex exponentials to simplify the Laplace transform of hyperbolic functions?
5. How do we apply partial fractions to simplify rational Laplace transforms?

**Tip:** The Laplace transform of hyperbolic functions is very similar to that of trigonometric functions. The key difference is the sign in the denominator, i.e., \( s^2 - a^2 \) instead of \( s^2 + a^2 \).

Learn how to find the Laplace transform of cosh^5(t) using properties of hyperbolic functions and Laplace transform formulas. This solution provides a detailed explanation of the steps involved in finding the transform, making it suitable for students studying differential equations and advanced calculus.

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