What is the General procedure for graphing heavidside functions?

Question

I was given an example of a second order differential equation with U1(t)-U(3t) as the forcing function. I was asked to graph the forcing function and the answer is that the function is 1 when t is between 1 and 3. I understand that heavidside functions are like a switch that turn on to be one. I thought that the interval would be one when t was less than one and 1 when t was greater than one. Is there a general procedure when asked to graph heaviside functions in a similar fashion to this problem? What if there was a constant infront of the heviside or what if they were added instead of subtracted?

Answer 1

(I'm understanding here that your $U_1(t)$ here is a Heaviside function that transitions at $t=1$.)

It's true to say that Heaviside functions are "like a switch" -- they start at one value, 0, and then they abruptly transition to the value of 1. You probably understand how to graph continuous functions, like $f(t) = t$, very well, and so only the discontinuous part makes it tricky, but you are correct that it changes at $t=1$ and $t=3$. If you can find these transition points, break the function up into parts where it is continuous, and graph those separately.

In this case, you correctly found that the transitions points are $1$ and $3$. In general, these will occur whenever the argument of $U_1$ passes through $1$. For instance, if I had $U_1(x^2-3)$ in my function, then my transition would be at $-2$ and $2$.

Take one point between each transition point, and consider what the function looks like there. Since $U_1$ is "locally constant", you can replace it with a constant as long as you're thinking about that interval. Here, your three intervals are $t < 1$, $1 < t < 3$, and $3 < t$. So just plug in values from each: evaluate your forcing function at $t=0$, $t=2$, and $t=4$. You know how $U_1$ will act with certain values plugged in, so then you can just see that you'll get $0, 1, 0$. Since it's constant there, that fully describes your function.

This procedure will also work for other things you mentioned, like adding, subtracting, leading constants, or other coefficients. But it also extends to many other things! If you want to plot a complicated function like $$f(t) = e^t U_0(-t) + t U_0(t) U_0(\sin t)$$ Notice that the first two $U$'s would transition when $t=0$, and the last transitions whenever $\sin t = 0$, that is, at $t = n \pi$. We want to see what happens on either side of $t=0$, so plug in $t = -1$ (only to the $U$s!), and we get $$f(t) = e^t * 1 + t * 0 * U(\sin t) = e^t$$ So we know that everywhere up to $t=0$, it's just an exponential function. Then we plug in $t=1$, and we get $$f(t) = e^t * 0 + t * 1 * U(\sin t) = t U(\sin t)$$ and this narrows it down. Then we know it's positive for $t$ after an even multiple of $\pi$, so it's just going to keep turning on and off there, with a steady ramp of $t$ the whole way.

I hope this illustrates how to tackle these kinds of problems in general!