Engineering Topics – Differential Equations

A differential equation is an equation that has a derivative in it. A derivative is a rate of change, like velocity. So if you are driving in a car where your velocity from a starting point is v which is some function of time, this can be solved to find your position from a starting point at any time. There are lots of techniques to solve these equations and the study of this scares many students. But the fact is, if you drive a car, pick up a glass of water, throw a ball at a target, your mind subconsciously handles the differential equations that model these activities quite well.

Let’s stick with the car example. Suppose you are 100 metres away from a stop sign or a red traffic light. So you apply the brakes. Let x(t) be your distance from the stop sign in metres at a time t, t be the time in seconds, and (t) be your velocity at time (t). Now you want your distance to the stop sign to decrease from 100 to 0 metres comfortably in say 15 seconds. What about this linear way:

It does stop at the stop sign, but is it comfortable? This equation of a line has a constant slope (rate of change) of −100/15 = −6.67 m/s = −24 km/hr. This means that at the stop sign, you are going 24 km/hr when you hit the brakes hard to stop. The passengers drinking coffee at the time, would not appreciate that. Also, if you are going 100 km/hr at 100 meters away, to follow this profile, you have to slam on the brakes to suddenly get at a speed of 24 km/hr. Well maybe there would be no coffee left to spill at the stop sign.

So this shows that we need to be aware of our speed and our distance to do this comfortably.

What about stopping following this red curve:

This starts at 100 metres and ends at 0 like the linear graph but has a better rate of change profile. The rate of change (that is the velocity) varies on this curve. It is visually seen as the slope of the line tangent to the graph at a point. The grey line shown is an example of a tangent line. Notice that the gradient at the beginning of the curve is high (in the negative direction) but at the end, it is near zero (the slope of a horizontal line is zero). This would be a much smoother stop than a linear approach.

But your mind during this action is not just seeing your distance from the stop sign, it is also sensing your velocity and adjusting it as you get closer to the stop sign. The following is an equation that relates the velocity and position:


If you were to solve this equation for x(t) using differential equation techniques, you would get the equation seen in the graph above. If you were to design a control system (which is what your mind is when performing this action) you would use the above differential equation to control both your position and your velocity.

But even this stopping profile has flaws. Notice that the deceleration at the beginning is quite steep (the slope of a tangent line at t = 0). Perhaps a better profile would be:

This starts with a more gentle deceleration, increases the deceleration until you get closer to the stop sign, then the deceleration decreases until you come to a full stop at the sign.

Regardless of the stopping profile used, your mind controls the braking action to conform to a desired profile based on your current speed (the slope of the tangent line) and your distance from the stop sign. People who are designing driverless cars, robotic arms, aircraft autopilots, etc, use differential equations. And because they are working in three dimensions, these equations can be in the form of matrix and/or vector equations. And the solutions will use complex numbers: all of these topics were covered in my last few posts.

So besides the basic algebraic skills you may be studying or have studied, more advanced topics like this one or those covered in in my last few post are the heart of engineering.