gatecse
asked
in Differential Equations
Feb 22, 2021
recategorized
Apr 11, 2021
by Lakshman Patel RJIT

0 votes

The Dirac-delta function $\left ( \delta \left ( t-t_{0} \right ) \right )$ for $\text{t}$, $t_{0} \in \mathbb{R}$, has the following property

$$\int_{a}^{b}\varphi \left ( t \right )\delta \left ( t-t_{0} \right )dt=\left\{\begin{matrix} \varphi \left ( t_{0} \right ) & a< t_{0}< b\\ 0 & \text{otherwise} \end{matrix}\right.$$

The Laplace transform of the Dirac-delta function $\delta \left ( t-a \right )$ for $a> 0$;

$\mathcal{L}\left ( \delta \left ( t-a \right ) \right )=F\left ( s \right )$ is

- $0$
- $\infty$
- $e^{sa}$
- $e^{-sa}$