Find
𝐿
{
7
𝑡
3
𝑒
−
5
𝑡
−
𝑡
2
+
cos
⁡
(
2
𝑡
)
}
.

Question

Find 
𝐿
{
7
𝑡
3
𝑒
−
5
𝑡
−
𝑡
2
+
cos
⁡
(
2
𝑡
)
}
 .

BlackTom AI · Accepted Answer

The provided question appears to ask for a Laplace transform, but the formatting is garbled and no answer options are given. I will first restate what can be inferred and then walk through the standard approach to compute the Laplace transform for a likely intended function, while clearly noting the missing pieces.

Assumed task (inference from the expression): Find L{ 7 t^3 e^{-5 t} - t^2 + cos(2 t) } = ? where L denotes the Laplace transform with respect to t.

Key Laplace transform facts that will be used:
- L{ t^n } = n! / s^{n+1} for n a nonnegative integer.
- L{ e^{−a t} f(t) } = F(s + a) if F(s) is the Laplace transform of f(t) (the Shifting Theorem).
- L{ cos(ω t) } = s / (s^2 + ω^2).

Step-by-step evaluation:
1) Break the function into parts: f(t) = 7 t^3 e^{−5 t} − t^2 + cos(2 t).
   We can compute the Laplace transform of each term separately and then sum the results due to linearity.

2) Compute L{ 7 t^3 e^{−5 t} }.
   - First compute L{ t^3 } = 3! / s^4 = 6 / s^4.
   - Apply the shifting theorem for e^{−5 t}: L{ t^3 e^{−5 t} } = 6 / (s + 5)^4.
   - Multiply by 7: L{ 7 t^3 e^{−5 t} } = 7 * 6 / (s + 5)^4 = 42 / (s + 5)^4.

3) Compute L{ − t^2 }.
   - L{ t^2 } = 2! / s^3 = 2 / s^3.
   - With the minus sign: L{ − t^2 } = − 2 / s^3.

4) Compute L{ cos(2 t) }.
   - Here ω = 2, so L{ cos(2 t) } = s / (s^2 + 4).

5) Combine the results using linearity:
   L{ 7 t^3 e^{−5 t} − t^2 + cos(2 t) } = 42 / (s + 5)^4 − 2 / s^3 + s / (s^2 + 4).

Notes on the missing data:
- The original input shows an empty array for answer_options, meaning there are no provided multiple-choice choices to evaluate. If options were supplied, we would compare the derived expression to each option, explaining which matches exactly and why the others do not.
- If the task intended a different interpretation of the expression, the same reasoning steps would adapt by using the corresponding transform rules for that function.

In summary, the Laplace transform of the given function, when interpreted as L{ 7 t^3 e^{−5 t} − t^2 + cos(2 t) }, is:
42 / (s + 5)^4 − 2 / s^3 + s / (s^2 + 4).

MAP2302 Lecture Quiz 19

Find 𝐿 { 7 𝑡 3 𝑒 − 5 𝑡 − 𝑡 2 + cos ⁡ ( 2 𝑡 ) } .

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