Find
𝐿
{
(
7
+
𝑒
−
7
𝑡
)
2
}
.

Question

Find 
𝐿
{
(
7
+
𝑒
−
7
𝑡
)
2
}
.

BlackTom AI · Accepted Answer

We start by restating the given problem clearly: Find L{ (7 + e − 7t)^2 }.
First, denote a = 7 + e, which is a constant since e is the base of natural logarithms and t is the variable. Then the expression inside the Laplace transform becomes (a − 7t)^2.
Next, expand the square: (a − 7t)^2 = a^2 − 14a t + 49 t^2.
Now apply the linearity of the Laplace transform term by term:
- L{a^2} = a^2 · L{1} = a^2 · (1/s) because the Laplace transform of 1 is 1/s.
- L{−14a t} = −14a · L{t} = −14a · (1/s^2) because the Laplace transform of t is 1/s^2.
- L{49 t^2} = 49 · L{t^2} = 49 · (2/s^3) because the Laplace transform of t^2 is 2/s^3.
Combine these terms:
L{(a − 7t)^2} = a^2/s − 14a/s^2 + 98/s^3.
Substitute back a = 7 + e, giving the final form:
L{(7 + e − 7t)^2} = (7 + e)^2 / s − 14(7 + e) / s^2 + 98 / s^3.
Note: the provided answer options are missing or empty in the prompt, so I cannot select a multiple-choice choice. If you supply the options, I can map this result to the correct one and explain why the others are incorrect.

MAP2302 Lecture Quiz 19

Find 𝐿 { ( 7 + 𝑒 − 7 𝑡 ) 2 } .

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