Find the laplace transform of cosht L{e^u coshudu

The Laplace Transform \( \mathcal{L}\{ f(t) \} \) of a function \( f(t) \) is given by the formula:

\[
\mathcal{L}\{ f(t) \} = \int_0^\infty e^{-st} f(t) \, dt
\]

Now, let's compute the Laplace Transform of the expression \( e^u \cosh(u) \). 

1. **Laplace Transform of \( \cosh(u) \)**:

   Recall the Laplace Transform of \( \cosh(at) \) is:

   \[
   \mathcal{L}\{ \cosh(at) \} = \frac{s}{s^2 - a^2}
   \]

   For \( \cosh(u) \), we have \( a = 1 \). So:

   \[
   \mathcal{L}\{ \cosh(u) \} = \frac{s}{s^2 - 1}
   \]

2. **Laplace Transform of \( e^u \cosh(u) \)**:

   Using the time-shifting property of the Laplace Transform, where if \( \mathcal{L}\{ f(t) \} = F(s) \), then \( \mathcal{L}\{ e^{at} f(t) \} = F(s - a) \), we can apply this to the Laplace Transform of \( e^u \cosh(u) \).

   Here, \( a = 1 \), and the Laplace Transform of \( \cosh(u) \) is \( \frac{s}{s^2 - 1} \). Therefore, the Laplace Transform of \( e^u \cosh(u) \) is:

   \[
   \mathcal{L}\{ e^u \cosh(u) \} = \frac{s - 1}{(s - 1)^2 - 1}
   \]

Simplifying the denominator:

\[
(s - 1)^2 - 1 = s^2 - 2s + 1 - 1 = s^2 - 2s
\]

Thus, the Laplace Transform of \( e^u \cosh(u) \) is:

\[
\mathcal{L}\{ e^u \cosh(u) \} = \frac{s - 1}{s^2 - 2s}
\]

Would you like more details or further clarification?

Here are 5 related questions to explore further:
1. What is the Laplace Transform of \( \sinh(u) \)?
2. How does the time-shifting property of Laplace Transforms work?
3. What is the inverse Laplace Transform of \( \frac{1}{s^2 - 1} \)?
4. Can we use Laplace Transforms to solve differential equations involving hyperbolic functions?
5. How does the Laplace Transform behave under multiplication by \( e^{at} \)?

**Tip**: The Laplace Transform is especially useful in solving differential equations, where it can convert derivatives into algebraic expressions.

Find the Laplace transform of cosht L{e^u coshudu

Learn how to compute the Laplace Transform of e^u cosh(u) using the time-shifting property. This problem explores key concepts such as Laplace transforms of hyperbolic functions and the time-shifting theorem. Suitable for undergraduate students studying calculus and differential equations.

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