what is the Laplace transform of e^4t(cos 2t) ?
To find the Laplace transform of e^4t(cos 2t), we will use the properties of the Laplace transform along with the formula for the Laplace transform of the cosine function. The formula for the Laplace transform of a function f(t) is given by:
L{f(t)} = ∫[0,∞] e^(-st) f(t) dt
where s is a complex variable.
We can use the following formula for the Laplace transform of the cosine function:
L{cos(ωt)} = s / (s^2 + ω^2)
In this case, f(t) = e^4t(cos 2t), so we need to find the Laplace transform of e^4t and multiply it by the Laplace transform of cos 2t.
1. Laplace transform of e^4t:
L{e^at} = 1 / (s - a), where a is a constant.
Substituting a = 4, we have:
L{e^4t} = 1 / (s - 4)
2. Laplace transform of cos 2t:
Using the formula mentioned earlier, with ω = 2, we have:
L{cos 2t} = s / (s^2 + 2^2) = s / (s^2 + 4)
Now, multiplying both Laplace transforms together:
L{e^4t(cos 2t)} = L{e^4t} * L{cos 2t}
= (1 / (s - 4)) * (s / (s^2 + 4))
= s / ((s - 4)(s^2 + 4))
Thus, the Laplace transform of e^4t(cos 2t) is s / ((s - 4)(s^2 + 4)).
To find the Laplace transform of the given function e^(4t) * cos(2t), we can use the properties of the Laplace transform along with the definition of the Laplace transform for elementary functions.
First, let's break down the given function into two parts - e^(4t) and cos(2t).
The Laplace transform of e^(4t) can be found using the formula:
L{e^(at)} = 1 / (s - a)
where "a" is the coefficient of the exponential term and "s" is the complex variable.
Applying this formula, we have:
L{e^(4t)} = 1 / (s - 4)
Next, let's find the Laplace transform of cos(2t). The Laplace transform of cos(2t) can be found using the formula:
L{cos(2t)} = s / (s^2 + a^2)
where "a" is the coefficient of the trigonometric function and "s" is the complex variable.
Applying this formula, we have:
L{cos(2t)} = s / (s^2 + 2^2) = s / (s^2 + 4)
Now, using the property of the Laplace transform for multiplication of functions, we can multiply the Laplace transforms of e^(4t) and cos(2t):
L{e^(4t) * cos(2t)} = L{e^(4t)} * L{cos(2t)}
Substituting the previously found Laplace transforms, we have:
L{e^(4t) * cos(2t)} = (1 / (s - 4)) * (s / (s^2 + 4))
Now, let's simplify this expression:
L{e^(4t) * cos(2t)} = (s / ((s - 4)(s^2 + 4)))
To finalize, we have derived the Laplace transform of the given function e^(4t) * cos(2t) as:
L{e^(4t) * cos(2t)} = (s / ((s - 4)(s^2 + 4)))
since there is e^4t multiplied to cos(2t), there is s-shifting:
from e^4t, we know that a = 4,, we also know that L{cos 2t} = s/(s^2 + 4) , therefore:
L{e^4t(cos 2t)} = (s-a)/((s-a)^2 + 4) = (s-4)/((s-4)^2 + 4)
so there,, =)