Find the Laplace transform of
𝐿
{
𝑡
2
cos
⁡
𝑏
𝑡
}
by using the formula

𝐿
{
𝑡
𝑛
𝑓
(
𝑡
)
}
(
𝑠
)
=
(
−
1
)
𝑛
𝑑
𝑛
𝑑
𝑠
𝑛
(
𝐿
{
𝐹
}
(
𝑠
)
)
,

Question

Find the Laplace transform of 
𝐿
{
𝑡
2
cos
⁡
𝑏
𝑡
}
by using the formula

𝐿
{
𝑡
𝑛
𝑓
(
𝑡
)
}
(
𝑠
)
=
(
−
1
)
𝑛
𝑑
𝑛
𝑑
𝑠
𝑛
(
𝐿
{
𝐹
}
(
𝑠
)
)
,

BlackTom AI · Accepted Answer

We start by restating the problem: find the Laplace transform of t^2 cos(bt) using the formula L{t^n f(t)}(s) = (-1)^n d^n/ds^n [L{f(t)}(s)].

First, compute the Laplace transform of cos(bt), which is a standard transform: L{cos(bt)}(s) = s/(s^2 + b^2).

Now apply the given formula with n = 2 and f(t) = cos(bt). Since (-1)^2 = 1, we need the second derivative of F(s) = s/(s^2 + b^2) with respect to s.

Step 1: Find F'(s). Using the quotient rule on F(s) = s/(s^2 + b^2):
- Numerator derivative: (s^2 + b^2) - s(2s) = s^2 + b^2 - 2s^2 = -s^2 + b^2.
- Denominator: (s^2 + b^2)^2.
Thus F'(s) = (-s^2 + b^2)/(s^2 + b^2)^2.

Step 2: Find F''(s), the derivative of F'(s). Let N(s) = -s^2 + b^2 and D(s) = (s^2 + b^2)^2. Then N'(s) = -2s and D'(s) = 2(s^2 + b^2)·(2s) = 4s(s^2 + b^2).
Using the quotient rule: F''(s) = [N'·D - N·D'] / D^2.
Compute the numerator:
N'·D = (-2s)(s^2 + b^2)^2,
N·D' = (-s^2 + b^2)[4s(s^2 + b^2)].
So the total numerator is (-2s)(s^2 + b^2)^2 - (-s^2 + b^2)[4s(s^2 + b^2)].
Factor (s^2 + b^2) to simplify: becomes [ -2s(s^2 + b^2) - 4s(-s^2 + b^2) ] · (s^2 + b^2) divided by (s^2 + b^2)^4, which simplifies to [ -2s(s^2 + b^2) - 4s(-s^2 + b^2) ] / (s^2 + b^2)^3.
Now simplify the bracketed expression:
-2s(s^2 + b^2) - 4s(-s^2 + b^2) = -2s^3 - 2sb^2 + 4s^3 - 4sb^2 = 2s^3 - 6sb^2 = 2s(s^2 - 3b^2).
Thus F''(s) = [2s(s^2 - 3b^2)] / (s^2 + b^2)^3.

Therefore, by the formula L{t^2 cos(bt)}(s) = F''(s) (since (-1)^2 = 1), we obtain:
L{t^2 cos(bt)}(s) = 2s(s^2 - 3b^2) / (s^2 + b^2)^3.

Note about the provided options: the question data shows an answer field and an answer_options field, but the answer_options list is empty. Consequently, there are no distinct choices to analyze individually. If options were provided, we would map each to a reasoning path: verify matches the derived expression, and explain why any incorrect form (e.g., missing factors, different powers, sign errors) would be invalid.

MAP2302 L19 section 7.3

Find the Laplace transform of 𝐿 { 𝑡 2 cos ⁡ 𝑏 𝑡 } by using the formula 𝐿 { 𝑡 𝑛 𝑓 ( 𝑡 ) } ( 𝑠 ) = ( − 1 ) 𝑛 𝑑 𝑛 𝑑 𝑠 𝑛 ( 𝐿 { 𝐹 } ( 𝑠 ) ) ,

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