In attempting to get the differential equation given a transfer function, I need to take through the third derivative of $r(t) = 5t^2 \, u(t)$, where $u(t)$ is the Heaviside function. If I "ignore" the $u(t)$, I can easily get the derivatives ($r'(t)=10 \, t \, u(t)$, $r''(t) = 10 \, u(t)$, $r'''(t)=0$). However, if I consider the $u(t)$, my first derivative becomes $r'(t)=5 \, t^2 \, \delta(t) + 10 \, t \, u(t)$, where $\delta(t)$ is the Dirac Delta Function.
Do I need to consider $u(t)$ when taking the derivative? If so, then what is the derivative of the Dirac Delta Function that I would use to get the second derivative?
Thanks for the help!
If it's a Heaviside of $u(t) = 0 $ at $ t < 0 $ and $ u(t) = 1 $ at $ t >= 0 $ then it seems to only make physical sense if you ignore the discontinuity below $ t = 0 $ . In effect this means ignoring the terms with $ u'(t), u''(t) ... $ etc.