The Dirac Delta Function could be applied to simplify the differential equation. There are three main properties of Dirac Delta Function.
$$\delta (x-x’) =\lim_{\tau\to0}\delta (x-x’)$$
such that,
$$ \delta (x-x’) = \begin{cases} \infty & x= x’ \ 0 & x\neq x’ \end{cases} $$
$$\int_{-\infty}^{\infty} \delta (x-x’)\ dx =1$$
Three Properties:
- Property 1:
$$\delta(x-x’)=0 \quad \quad ,x\neq x’ $$
- Property 2:
$$ \int_{x’-\epsilon}^{x’+\epsilon} \delta (x-x’)dx =1\quad \quad ,\epsilon >0 $$
- Property 3:
$$\int_{x’-\epsilon}^{x’+\epsilon} f(x)\ \delta (x-x’)dx = f(x’)$$
At x=x’ the Dirac Delta function is sometimes thought of has having an “infinite” value. So, the Dirac Delta function is a function that is zero everywhere except one point and at that point it can be thought of as either undefined or as having an “infinite” value.