evaluate the laplace transform of the function f(t)={0 t<3 (t-3)^3 t>=3

To evaluate the Laplace transform of the function \( f(t) \), which is defined as

\[
f(t) = 
\begin{cases} 
0, & t < 3 \\
(t - 3)^3, & t \geq 3 
\end{cases}
\]

we can approach this using the **Laplace transform for shifted functions**.

### Step 1: Recognize the Shift
Since \( f(t) = (t - 3)^3 \) for \( t \geq 3 \), this suggests a **shifted function**. The Heaviside (unit step) function \( u(t - 3) \) can be used to rewrite \( f(t) \) as:
\[
f(t) = (t - 3)^3 u(t - 3)
\]

### Step 2: Use the Laplace Transform of a Shifted Function
The Laplace transform of a function \( g(t - a) u(t - a) \) is given by:
\[
\mathcal{L}\{g(t - a) u(t - a)\} = e^{-as} G(s)
\]
where \( G(s) \) is the Laplace transform of \( g(t) \) (i.e., \( G(s) = \mathcal{L}\{g(t)\} \)).

In this case, \( g(t) = t^3 \) and \( a = 3 \).

### Step 3: Find the Laplace Transform of \( g(t) = t^3 \)
The Laplace transform of \( t^3 \) is:
\[
\mathcal{L}\{t^3\} = \frac{6}{s^4}
\]

### Step 4: Apply the Shift
Now, using the shift property, we have:
\[
\mathcal{L}\{(t - 3)^3 u(t - 3)\} = e^{-3s} \cdot \frac{6}{s^4}
\]

### Final Answer
Therefore, the Laplace transform of \( f(t) \) is:
\[
\mathcal{L}\{f(t)\} = \frac{6 e^{-3s}}{s^4}
\]

Would you like further details or have any questions?

---

Here are some related questions you might find useful:

1. What is the Laplace transform of other shifted functions?
2. How does the Heaviside function simplify the transformation of piecewise functions?
3. How would this approach change if the exponent of \( t \) was different?
4. What happens if the shift occurs at \( t = 0 \) rather than \( t = 3 \)?
5. How would you take the inverse Laplace transform of similar expressions?

**Tip:** Remember, the Heaviside (unit step) function is a useful tool for expressing and transforming piecewise functions in Laplace transforms.

Explore the evaluation of the Laplace transform for the piecewise function f(t) defined as 0 for t < 3 and (t-3)^3 for t >= 3. This solution covers the use of the Heaviside function and the shift theorem, making it ideal for advanced high school or college-level 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