To find the Laplace transform of the given function \( t^2 e^{-3t} \sin(21t) \), we can apply the following steps.

### Step 1: Recognize the function form
The function can be expressed as:
\[
f(t) = t^2 e^{-3t} \sin(21t)
\]
We can identify that we need to handle both the polynomial term \( t^2 \), the exponential decay \( e^{-3t} \), and the sine term \( \sin(21t) \).

### Step 2: Use the shifting property of the Laplace transform
The presence of \( e^{-3t} \) suggests we will use the **shifting theorem**:
\[
\mathcal{L}\{ e^{at} f(t) \} = F(s - a)
\]
For \( e^{-3t} \), this becomes a shift by 3 to the right. So we first find the Laplace transform of \( t^2 \sin(21t) \), and then apply the shift.

### Step 3: Laplace transform of \( t^2 \sin(21t) \)
The Laplace transform of \( t^n \sin(bt) \) is given by:
\[
\mathcal{L}\{ t^n \sin(bt) \} = \frac{n! \cdot b}{(s^2 + b^2)^{n+1}}
\]
For \( t^2 \sin(21t) \), we have:
\[
\mathcal{L}\{ t^2 \sin(21t) \} = \frac{2! \cdot 21}{(s^2 + 21^2)^3} = \frac{42}{(s^2 + 441)^3}
\]

### Step 4: Apply the shifting theorem
Now, we apply the shifting property due to \( e^{-3t} \):
\[
\mathcal{L}\{ e^{-3t} t^2 \sin(21t) \} = \frac{42}{((s + 3)^2 + 441)^3}
\]

Thus, the Laplace transform of \( t^2 e^{-3t} \sin(21t) \) is:
\[
\mathcal{L}\{ t^2 e^{-3t} \sin(21t) \} = \frac{42}{((s + 3)^2 + 441)^3}
\]

Let me know if you'd like any further explanation or have questions on this process!

### Related Questions:
1. What is the Laplace transform of \( e^{-at} \cos(bt) \)?
2. How does the Laplace transform of \( t^n \sin(bt) \) change for different values of \( n \)?
3. Can we apply the shifting property to functions other than exponentials?
4. How does the Laplace transform handle higher powers of \( t \) (like \( t^3 \))?
5. What is the inverse Laplace transform of \( \frac{s}{s^2 + \omega^2} \)?

**Tip**: Always check if a function has an exponential term, as the shifting property simplifies the Laplace transform significantly!

Explore how to compute the Laplace transform of the function t^2 e^-3t sin(21t). This detailed solution covers the use of the shifting theorem and key formulas for handling polynomial and trigonometric functions. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation