Question

Laplace Transform

Answer 1

Banach space

your turn ;)

Answer 2

Use the definition of Laplace transform to find
$$\mathcal{L}\{f(t)\}(s)$$, where $$f(t)=\left\{\begin{array}{cc}
0 & 0\leq t< 4 \\
2+t & t\geq 4
\end{array}\right.$$
(be sure to
include any restrictions on s)

$$\mathcal{L}\{f(t)\}(s)=\int_{0}^{\infty}e^{-st}f(t)dt$$

$$=\int_{4}^{\infty}e^{-st}(2+t)dt=\left.\left(\frac{-t}{s}-\frac{2}{s}-\frac{1}{s^2}\right)e^{-ts}\right|_{4}^{\infty}$$

$$=\frac{(6s+1)e^{-4s}}{s^2}$$

$$s>0$$