Dear Vijayalakshmi Palanisamy,
editing a notebook is definitely not the responsibility of this site! But maybe I can make some suggestions: When you have a differential equation with a time delay you can rewrite this into a "regular" differential equation by replacing the delayed terms by its series expansion:
$
x(t-\tau)=x(-\tau )+t x'(-\tau )+\frac{1}{2} t^2 x''(-\tau )+\frac{1}{6} t^3 x^{(3)}(-\tau ) + \cdots
$
This is as trivial as fundamental, and it shows the true nature of the problem: Time delay makes it a differential equation of (theoretically) infinite order.
What might help is to make an integral transform (e.g. FourierTransform) of the equations:
Concerning delay you get:
$
f(t-\tau) \longrightarrow F(\kappa)\mbox{e}^{-\imath 2\pi\kappa \tau}
$
and any fractional order can also quite nicely be expressed because we always have (for any
$n$):
$
f^{(n)}(t) \longrightarrow (+2\pi\imath\kappa)^n F(\kappa)
$
OK, these are just some thoughts from a non(!)-mathematician.
Regards -- Henrik
