L((t−3)2u(t−2)L((t−3)2u(t−2)


a.

(1s2−2s2+1s)e−2s(1s2−2s2+1s)e−2s


b.

(2!s2+2s2+1s)e−2s(2!s2+2s2+1s)e−2s

c.

None

d.

(2!s2−2s2+1s)e−2s


To find the Laplace transform of the function \((t - 3)^2 u(t - 2)\), we can proceed as follows:

### Step 1: Understand the Function
- The function is \((t - 3)^2 u(t - 2)\).
- Here, \( u(t - 2) \) is the unit step function that shifts the function to the right by 2 units.

### Step 2: Apply the Shift Theorem
The time shift theorem for the Laplace transform states:

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

For the given function, we can rewrite \((t - 3)^2 u(t - 2)\) as:

\[
(t - 3)^2 u(t - 2) = \left[(t - 2) + (2 - 3)\right]^2 u(t - 2) = (t - 2 - 1)^2 u(t - 2)
\]

However, to make it simpler, notice that \((t - 3)\) is just a shift of \( (t - 2) - 1 \), and so it's easier to deal with it by directly applying the Laplace transform to \( (t - 2)^2 u(t - 2) \).

### Step 3: Find the Laplace Transform
Now, let's find the Laplace transform of \( (t - 2)^2 \):

\[
\mathcal{L}\{ (t - 2)^2 u(t - 2) \} = e^{-2s} \mathcal{L}\{ t^2 \}
\]

The Laplace transform of \( t^2 \) is:

\[
\mathcal{L}\{ t^2 \} = \frac{2!}{s^3} = \frac{2}{s^3}
\]

So, the Laplace transform of the entire function is:

\[
\mathcal{L}\{ (t - 3)^2 u(t - 2) \} = e^{-2s} \frac{2}{s^3}
\]

None of the given options seem to match exactly. Hence, the correct answer is:

**c. None**

Would you like further details or have any questions? Here are some related questions:

1. What is the Laplace transform of \( t^n \) for any integer \( n \)?
2. How does the unit step function \( u(t - a) \) affect the Laplace transform?
3. Can you find the Laplace transform of \( e^{at} \cos(bt) u(t - c) \)?
4. How does the time-shifting property work in Laplace transforms?
5. What are some common applications of the Laplace transform in engineering?

**Tip:** When dealing with the unit step function \( u(t - a) \), remember to apply the time-shifting property of the Laplace transform, which introduces an exponential factor \( e^{-as} \).

Learn how to find the Laplace transform of the function (t - 3)^2 u(t - 2) using the time shift theorem and Laplace transform properties. This problem involves advanced concepts in Laplace transforms and unit step functions.

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