The second condition for linearity given above is a condition not on the relationship between the parts of the system, but on the relationship between the quantities described by the differential equation in question. (2) demands that the way that the quantities described by the equation vary with respect to one another remain constant. To get a sense of what that means, it’s probably easiest to think about some cases where the requirement holds, and then think about some cases where the requirement doesn’t hold. Suppose you’re walking on a treadmill, and want to vary the speed at which the belt is moving so that you walk more quickly or more slowly. You can do this by pressing the up and down arrows on the speed control; each time you press one of the arrows, the speed of the belt will change by (say) .1 MPH. This is an example of a variation that satisfies condition (2). We could write down a simple differential equation relating two quantities: the number of times you’ve pressed each button, and the speed at which the treadmill’s belt is moving. No matter how many times you press the button, though, the value of the button press will remain constant: the amount by which pressing the up arrow varies the speed doesn’t depend on how many times you’ve pressed the button, or on how fast the treadmill is already turning. Whether you’re walking slowly at one mile per hour or sprinting at 15 miles per hour, pressing that button will always result in a change of .1 mile per hour. Condition (2) is satisfied.
OK, with an understanding of what a system must look like in order to be linear, let’s think about what sorts of systems might fail to satisfy these requirements. Let’s return to the treadmill
- Actually, this case satisfies both conditions. We’ve just seen how it satisfies (2), but we could also break the system apart and consider your “up arrow” presses and “down arrow” presses independently of one another and still calculate the speed of the belt. Treadmill speed control is a linear system, and this underscores the point that conditions (1) and (2) are not as independent as this presentation suggests.