To solve for $f'(t) = 3e^{2t} + 4\sin(3t)$ using the first derivative property of the Laplace transform, we will proceed step by step.

Step 1: Recall the first derivative property of the Laplace transform

The first derivative property states that:

$\mathcal{L} \{ f'(t) \} = sF(s) - f(0)$

where $F(s)$ is the Laplace transform of $f(t)$, and $f(0)$ is the initial condition of $f(t)$ at $t = 0$.

Step 2: Express $f'(t)$

Given: $f'(t) = 3e^{2t} + 4\sin(3t)$

We need to take the Laplace transform of each term.

(a) Laplace Transform of $3e^{2t}$

Using the Laplace transform of $e^{at}$, which is $\mathcal{L} \{ e^{at} \} = \frac{1}{s - a}$, we get:

$\mathcal{L} \{ 3e^{2t} \} = 3 \cdot \frac{1}{s - 2} = \frac{3}{s - 2}$

(b) Laplace Transform of $4\sin(3t)$

Using the Laplace transform of $\sin(at)$, which is $\mathcal{L} \{ \sin(at) \} = \frac{a}{s^2 + a^2}$, we get:

$\mathcal{L} \{ 4\sin(3t) \} = 4 \cdot \frac{3}{s^2 + 9} = \frac{12}{s^2 + 9}$

Step 3: Add the Laplace transforms

Now, the Laplace transform of $f'(t)$ is the sum of the transforms:

$\mathcal{L} \{ f'(t) \} = \frac{3}{s - 2} + \frac{12}{s^2 + 9}$

Step 4: Use the first derivative property to find $\mathcal{L} \{ f(t) \}$

Using the property $\mathcal{L} \{ f'(t) \} = sF(s) - f(0)$, and assuming $f(0) = 0$ (if no initial condition is given), we have:

$sF(s) = \frac{3}{s - 2} + \frac{12}{s^2 + 9}$

Thus:

$F(s) = \frac{\frac{3}{s - 2} + \frac{12}{s^2 + 9}}{s}$

This expression represents the Laplace transform of $f(t)$.

Conclusion

Using the first derivative property of the Laplace transform, we found that the Laplace transform of the original function $f(t)$ is:

$F(s) = \frac{3}{s(s - 2)} + \frac{12}{s(s^2 + 9)}$

If you'd like to go further and invert the transform to find $f(t)$, or need more details on any step, let me know!

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Do you want to see the details of the inverse transform or have any specific questions?

Here are some related problems you could explore:

1. Find the inverse Laplace transform of $\frac{3}{s(s - 2)}$.
2. Use the Laplace transform to solve $f''(t) = 3e^{2t} + 4\sin(3t)$ with $f(0) = 0$, $f'(0) = 1$.
3. Solve a second-order differential equation using Laplace transforms.
4. Explore the Laplace transform of $e^{at} \cos(bt)$.
5. How does the second derivative property of the Laplace transform work?

Tip: The Laplace transform is particularly useful for solving differential equations with initial conditions, as it simplifies them into algebraic equations.