find the laplace transform of [e^-4t/3 sin(3t)]

The Laplace transform of the given function \( f(t) = e^{-\frac{4t}{3}} \sin(3t) \) can be computed using the known Laplace transform formulas for exponential and sinusoidal functions.

The general formula for the Laplace transform of a product of an exponential function and a sinusoidal function is:

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

In this case:
- \( a = -\frac{4}{3} \)
- \( b = 3 \)

So, applying the formula:

\[
\mathcal{L}\{ e^{-\frac{4t}{3}} \sin(3t) \} = \frac{3}{\left( s + \frac{4}{3} \right)^2 + 3^2}
\]

Now, simplify the denominator:

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

Thus, the Laplace transform of \( e^{-\frac{4t}{3}} \sin(3t) \) is:

\[
\boxed{\frac{3}{\left( s + \frac{4}{3} \right)^2 + 9}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you compute the Laplace transform of \( e^{-2t} \sin(5t) \)?
2. What is the inverse Laplace transform of \( \frac{4}{(s+1)^2 + 16} \)?
3. Can you apply the same method to find the Laplace transform of \( e^{2t} \cos(4t) \)?
4. How does the shift theorem apply in Laplace transforms?
5. How do you compute the Laplace transform of \( t e^{-t} \sin(2t) \)?

**Tip:** The Laplace transform is especially useful for solving differential equations by converting them into algebraic equations.

This page provides a detailed solution to finding the Laplace transform of e^(-4t/3) sin(3t), using the standard formulas for Laplace transforms of exponential and sinusoidal functions. Learn how to apply the Laplace transform theorem and the exponential shift rule. Suitable for undergraduate students.

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