Ch3_Differential_Equations - Chapter 3: Differential...

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Chapter 3: Differential Equations – Expressing things in terms of change Scott Owen & Greg Corrado The first piece of good news is that differential equations are fairly straightforward in concept. Any of you who have actually taken a class in differential equations will be cringing at this point, having learned from bitter experience that solving differential equations is often a tricky matter. But fear not. Writing a differential equation to describe your own favorite dynamic system is often a straightforward process, and that’s the science comes in. Where much of the confusion and frustration comes is in generating a solution to that equation, or otherwise extracting information from the equation once it is written. The good news is that we don’t care. For a neuroscientist or psychologist the approximate numerical solutions provided by software such as Mathematica, Matlab, and even (in a pinch) Microsoft Excel are sufficiently accurate in nearly every case. An aerospace engineer might demand higher standards of accuracy (but they have wind tunnels for a reason). In our business there’s little point to predicting something to 6 significant digits of precision when you can only measure it to 2. This chapter will provide a definition of differential equations and, through examples, introduce a few of the forms of differential equations what their solutions tend to look like. Definition: A differential equation is any equation that describes a function in terms of its own derivative. The most basic differential equation you can imagine writing is: d dt f ( t ) = m This equation says that derivative of the function, f ( t ) , is equal to a constant, m . A function that has a constant derivative? That’s just a function that has a steady, perfectly uniform slope. Well a straight line has a constant slope right. So what if f ( t ) = m ! t + b ? If you try taking the derivative of that, you’ll see immediately that it satisfy our equation because the derivative of that function is m , plain and simple. (This ‘guess and check’ strategy may seem hokey, but perhaps terrifyingly it’s often how differential equations are solved.) But what about this b variable – where did that come from, what value does b need to have in order to satisfy our equation? It shouldn’t take you long to realize that any value of b would fit; it could be zero, it could be a billion, our differential equation doesn’t tell us. We haven’t done anything wrong; this happens all the times with differential equations. A differential equation describes a class of systems, and so always has many solutions. But the solutions all come from a certain family, in this case the family is the set of all lines with slope m . To figure out which of these solutions fits your particular situation (for example if you’re trying to predict how some physical system governed by this differential equation works), then you need some additional information. Engineers tend to cal this side information the “initial conditions” or sometimes the
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Ch3_Differential_Equations - Chapter 3: Differential...

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