To find the Laplace transform of the product \(\sinh(2t) \sin(3t)\), we can use the formula for the Laplace transform of a product of hyperbolic and trigonometric functions:

\[
\mathcal{L} \{ \sinh(at) \sin(bt) \} = \frac{ab}{(s^2 - a^2)(s^2 + b^2)}
\]

Given \(a = 2\) and \(b = 3\), let's plug these values into the formula:

\[
\mathcal{L} \{ \sinh(2t) \sin(3t) \} = \frac{2 \cdot 3}{(s^2 - 2^2)(s^2 + 3^2)}
\]

Simplifying the expression:

\[
\mathcal{L} \{ \sinh(2t) \sin(3t) \} = \frac{6}{(s^2 - 4)(s^2 + 9)}
\]

So, the Laplace transform of \(\sinh(2t) \sin(3t)\) is:

\[
\boxed{\frac{6}{(s^2 - 4)(s^2 + 9)}}
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the Laplace transform of \(\cosh(2t) \sin(3t)\)?
2. How can we compute the inverse Laplace transform of \(\frac{6}{(s^2 - 4)(s^2 + 9)}\)?
3. What is the Laplace transform of \(\sinh(2t) \cos(3t)\)?
4. How to find the Laplace transform of \(\sinh(at) \cos(bt)\)?
5. What properties of the Laplace transform are useful for solving differential equations?

### Tip:
To solve problems involving Laplace transforms, familiarize yourself with the standard transform tables and properties like linearity, shifting, and the initial/final value theorems.

Discover how to compute the Laplace transform of sinh(2t) sin(3t) using the formula for products of hyperbolic and trigonometric functions. This detailed solution covers step-by-step calculation and simplification, suitable for advanced college-level students and professionals in mathematics and engineering.

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