https://file.aiquickdraw.com/mathbot/file/17298760937270bq2rmix.jpg

To find the Laplace transform of the given piecewise function:

\[
f(t) = 
\begin{cases}
0, & t < 2 \\
t^2 - 4t + 9, & t \geq 2
\end{cases}
\]

### Step-by-Step Approach:

Since the function \(f(t)\) is zero before \(t = 2\), we need to shift and handle the function using the **Heaviside (unit step) function**. 

We can rewrite the given function in terms of the Heaviside function \(u(t - 2)\), which activates the non-zero part of the function at \(t = 2\):

\[
f(t) = \left(t^2 - 4t + 9\right) u(t - 2).
\]

Now, we use the **time-shifting property of the Laplace transform**:

### Time-Shifting Property:
If \(f(t) = g(t - a) u(t - a)\), then:

\[
\mathcal{L}\{f(t)\} = e^{-as} \mathcal{L}\{g(t)\}.
\]

In our case, \(a = 2\) and \(g(t) = t^2 - 4t + 9\). However, since \(t\) shifts to \(t - 2\) under the Heaviside function, we must first replace \(t\) with \((t + 2)\) in the function \(t^2 - 4t + 9\).

### Step 1: Replace \(t\) with \(t + 2\)
The expression becomes:

\[
(t + 2)^2 - 4(t + 2) + 9.
\]

Simplifying:

\[
t^2 + 4t + 4 - 4t - 8 + 9 = t^2 + 5.
\]

Thus, \(g(t) = t^2 + 5\).

### Step 2: Find the Laplace Transform of \(t^2 + 5\)
Using standard Laplace transform rules:

\[
\mathcal{L}\{t^2\} = \frac{2}{s^3}, \quad \mathcal{L}\{5\} = \frac{5}{s}.
\]

Thus:

\[
\mathcal{L}\{t^2 + 5\} = \frac{2}{s^3} + \frac{5}{s}.
\]

### Step 3: Apply the Time-Shift Property
Using the time-shift property:

\[
\mathcal{L}\{(t^2 - 4t + 9) u(t - 2)\} = e^{-2s} \left(\frac{2}{s^3} + \frac{5}{s}\right).
\]

### Final Answer:
The Laplace transform of the given function is:

\[
\boxed{e^{-2s} \left(\frac{2}{s^3} + \frac{5}{s}\right)}.
\]

Let me know if you want further clarification or details on any step! 

---

Here are 5 related questions:
1. How do you derive the Laplace transform of a polynomial function?
2. What is the role of the Heaviside function in Laplace transforms?
3. How does the time-shifting property of Laplace transforms work?
4. How can Laplace transforms be applied to solve differential equations?
5. What are common mistakes when working with piecewise functions in Laplace transforms?

**Tip:** Always remember to adjust the limits and variables properly when working with the Heaviside function in Laplace transforms.

Find the Laplace transform of f(t) = {0, t < 2; t^2 - 4t + 9, t >= 2}

Find the Laplace transform of the piecewise function f(t) = {0, t < 2; t^2 - 4t + 9, t >= 2} using the Heaviside function and time-shifting property. This step-by-step solution explains the application of Laplace transforms and the key formulas involved.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation