Your Question:

2. Using the Laplace transform pairs of Table 2.1 and the Laplace transform theroems of Table 2.2,

derive the Laplace transforms for the following time functions:

c. 𝑡3 cos(𝜔𝑡) 𝑢(𝑡)

Step By Step Answers with Explanation

To derive the Laplace transforms for the given time functions using Laplace transform pairs and Laplace transform theorems, we can follow these steps:

\[ \mathcal{L}\{\sin(\omega t)u(t)\} = \frac{\omega}{s^2 + \omega^2} \]

3. Laplace Transform of \(\cos(\omega t) u(t)\):

\[ e^{-at} \sin(\omega t) u(t) = \frac{1}{2i} \left[\mathcal{L}\{e^{at}\} \mathcal{L}\{\sin(\omega t) u(t)\}\right] \]

Now, apply the Laplace transforms and convolution:

\[ \frac{1}{2i} \int_0^t \frac{1}{s - a} \cdot \frac{\omega}{s^2 + \omega^2} ds \]

After evaluating this integral, you'll obtain the Laplace transform for the function.

To find the Laplace transform of \(t^3 \cos(\omega t) u(t)\), you can use the differentiation theorem, which states that if the Laplace transform of \(f(t)\) is \(F(s)\), then the Laplace transform of \(t^n f(t)\) is \((-1)^n\) times the nth derivative of \(F(s)\) with respect to \(s\).

So, the Laplace transform of \(t^3 \cos(\omega t) u(t)\) is: