Suppose we model a physical phenomenon with a 2nd order linear differential equation:
$a_2$(t)$y''$+$a_1$(t)$y'$+$a_0$(t)$y$=$f(t)$,
where 't' stands for time.
- In choosing an appropriate driving function $f(t)$, suppose we want just the segment of a certain function $g(t)$ for 5≤t<10 only.
- Further, suppose we want this segment to be active for the duration 15≤t<20
- while for 0≤t<15 and 20≤t, we want f(t)=0
describe this input function $f(t)$ using the unit step function.
I think it's just how wordy this question is, but I am completely lost, any help or even just a step in the right direction is greatly appreciated. Thanks again guys.
A step (heaviside) function is defined $$u(t)=\begin{cases}0&t<0\\1&t<0\end{cases}$$ To shift the jump right to $t=\alpha>0$ units one need $u(t-\alpha)$. Similarly to shift it left one need $u(t+\alpha)$. To shift it up/down one need $u(t)\pm\alpha$. To scale it one need $\alpha u(t)$. Finally to switch it upside down one need $-u(t)$.
For a range $a\leq t\leq b$ of $g(t)$ you will need $$g(t)u(t-a) - g(t)u(t-b)= \begin{cases}0&t<a\\ g(t) & a \leq t \leq b\\ 0 & t\ge b \end{cases}$$
In the last one you want shifted $g(t)$ at $[a,b]$, i.e. you need $g(t-\alpha)$ instead of $g(t)$. Note you have $a=15$, $b=20$, $\alpha =10$. Finally $$g(t-10)u(t-15) - g(t-10)u(t-20)$$