Question

Write each function in terms of unit step functions. Find the Laplace transform of the given function.

Short answer

Laplace transform of the given function is$Y\left(s\right)=\frac{e^{-as}}{s}-\frac{e^{-bs}}{s}$

Step-by-step solution

Definition of Laplace transform

A Laplace transform of $f\left(t\right)$function in a time domain, whereis the real number greater than or equal to zero, is given as$F\left(s\right)$where there

$F\left(s\right)={\int }_0^{\gamma }f\left(t\right)e^{-st}dt$

S is the complex frequency domain.

Step 2:Find the Laplace transform of the above function

Representing the given graph as a unit step "switch" function:

$y\left(t\right)=U\left(t-a\right)-U\left(t-b\right)$

The unit step function is defined to be,

$Theunitstepfunction9\left(t-a\right)isdefinedtobe,$

$J\left(t-a\right)=\left\{\begin{array}{ll}0,& 0⩽t<a\\ 1& t\ge a\end{array}\right.$

$L\left\{y\left(t\right)\right\}=L\left\{U\left(t-a\right)\right\}-L\left\{U\left(t-b\right)\right\}$

Hence, the Laplace transform of the above function is$Y\left(s\right)=\frac{e^{-as}}{s}-\frac{e^{-bs}}{s}$