Answer to Question #239730 in Differential Equations for sam

Question

 Find the Laplace transform, if it exists, of each of the following functions

(a) f(t) = e^at

(b) f(t) = 1

(c) f(t) = t 

Answer 1

(a)

$$F(s)=L(f(t))=L(e^{at})=\displaystyle\int_{0}^{\infin}e^{at}e^{-st}dt$$

$$=-\dfrac{1}{s-a}\lim\limits_{A\to\infin}[e^{(a-s)t}]\begin{matrix} A \\ 0 \end{matrix}=-\dfrac{1}{s-a}(0-1)$$

$$=\dfrac{1}{s-a}$$

(b)

$$F(s)=L(f(t))=L(1)=\displaystyle\int_{0}^{\infin}(1)e^{-st}dt$$

$$=-\dfrac{1}{s}\lim\limits_{A\to\infin}[e^{-st}]\begin{matrix} A \\ 0 \end{matrix}=-\dfrac{1}{s}(0-1)$$

$$=\dfrac{1}{s}$$

(c)

$$F(s)=L(f(t))=L(t)=\displaystyle\int_{0}^{\infin}(t)e^{-st}dt$$

$$=-\dfrac{1}{s}\lim\limits_{A\to\infin}[te^{-st}+\dfrac{1}{s}e^{-st}]\begin{matrix} A \\ 0 \end{matrix}=-\dfrac{1}{s}(0-0+0-\dfrac{1}{s})$$

$$=\dfrac{1}{s^2}$$