find the Laplace transform of f(t)=e^-2t sin (4t) using the time scaling domain method

To find the Laplace transform of the function \( f(t) = e^{-2t} \sin(4t) \) using the **time-scaling method**, we need to recognize that this function is a product of an exponential function and a sine function. The standard form of the Laplace transform can be applied as follows:

### Step-by-Step Solution:

1. **Identify the formula for the Laplace transform**:
   The Laplace transform of \( e^{at} f(t) \) is \( \mathcal{L}\{ e^{at} f(t) \}(s) = F(s - a) \), where \( F(s) \) is the Laplace transform of \( f(t) \).

2. **Use the Laplace transform of \( \sin(bt) \)**:
   The Laplace transform of \( \sin(bt) \) is:
   \[
   \mathcal{L}\{ \sin(bt) \}(s) = \frac{b}{s^2 + b^2}
   \]
   In this case, \( b = 4 \), so the Laplace transform of \( \sin(4t) \) is:
   \[
   \mathcal{L}\{ \sin(4t) \}(s) = \frac{4}{s^2 + 16}
   \]

3. **Apply the time-scaling property**:
   Now apply the time-scaling property, which accounts for the exponential term \( e^{-2t} \). This shifts the \( s \)-domain by \( -a \), where \( a = -2 \). The formula is:
   \[
   \mathcal{L}\{ e^{-2t} \sin(4t) \}(s) = \frac{4}{(s + 2)^2 + 16}
   \]

4. **Final Result**:
   Therefore, the Laplace transform of \( f(t) = e^{-2t} \sin(4t) \) is:
   \[
   \mathcal{L}\{ e^{-2t} \sin(4t) \}(s) = \frac{4}{(s + 2)^2 + 16}
   \]

### Do you have any questions or need further details?

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Here are 5 related questions that build on this concept:

1. What is the Laplace transform of \( e^{3t} \cos(5t) \)?
2. How would you find the inverse Laplace transform of \( \frac{6}{(s - 3)^2 + 25} \)?
3. What is the effect of shifting the function \( f(t) \) on its Laplace transform?
4. How do you apply the Laplace transform to differential equations involving sinusoidal functions?
5. What happens when both \( e^{at} \) and \( \cos(bt) \) are part of the function being transformed?

**Tip**: When using the Laplace transform for sinusoidal functions, always remember that exponential shifts translate to shifts in the \( s \)-domain!

Explore how to find the Laplace transform of the function e^-2t sin(4t) using the time-scaling method. This detailed step-by-step solution covers the Laplace transform formulas for exponential and sine functions and applies the time-scaling property. Suitable for college-level students studying Laplace transforms.

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