Question

find f(s).
f(s) =
l{te^{14t}}

Explanation:

Step1: Recall Laplace Transform Rule

We use the Laplace transform rule for \( \mathcal{L}\{t^n e^{at}\} \), where \( n = 1 \) and \( a = 14 \). The formula is \( \mathcal{L}\{t^n e^{at}\}=\frac{n!}{(s - a)^{n + 1}} \) for \( s>a \).

Step2: Substitute \( n = 1 \) and \( a = 14 \)

For \( n = 1 \), \( n! = 1! = 1 \). Substituting into the formula, we get \( \mathcal{L}\{t e^{14t}\}=\frac{1!}{(s - 14)^{1+1}}=\frac{1}{(s - 14)^2} \) (for \( s>14 \)).

Answer:

\( \frac{1}{(s - 14)^2} \)