
Find the Laplace transform of ( f(t) = e^{2t} \sin(3t) ) by first shifting

To find the Laplace transform of \( f(t) = e^{2t} \sin(3t) \), we can use the **Laplace transform shifting theorem**. The shifting theorem states that if:

\[
\mathcal{L}\{e^{at} f(t)\}(s) = F(s-a)
\]

We will start by finding the Laplace transform of \( \sin(3t) \) and then apply the shifting rule.

### Step 1: Find the Laplace transform of \( \sin(3t) \)
The Laplace transform of \( \sin(bt) \) is:

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

For \( \sin(3t) \), we substitute \( b = 3 \):

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

### Step 2: Apply the shifting theorem
Now, using the shifting theorem for \( e^{2t} \sin(3t) \), we replace \( s \) with \( s - 2 \) in the Laplace transform of \( \sin(3t) \):

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

Thus, the Laplace transform of \( f(t) = e^{2t} \sin(3t) \) is:

\[
\mathcal{L}\{f(t)\}(s) = \frac{3}{(s - 2)^2 + 9}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you apply the Laplace transform to exponential functions?
2. What is the inverse Laplace transform of \( \frac{3}{(s - 2)^2 + 9} \)?
3. How does the shifting theorem work for different kinds of functions?
4. Can you find the Laplace transform of \( e^{3t} \cos(4t) \)?
5. How does the convolution theorem relate to the Laplace transform?

**Tip**: Always check for initial conditions when applying the Laplace transform to differential equations!

Find the Laplace transform of ( f(t) = e^{2t} \sin(3t) ) by first shifting

This guide explains how to find the Laplace transform of f(t) = e^{2t} sin(3t) by applying the shifting theorem. It covers key mathematical concepts, including the Laplace transform of trigonometric functions and the shifting rule. Ideal for undergraduate-level students studying differential equations.

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