MAT1193 – 10a Differential Equations -­‐ Definitions A differential equation is simply an equation with a derivative. In simplest types of differential equation, the left hand side is a derivative and the right hand side is a function. Examples are: a) b)
c)
Notice that simply from the notation for the derivative you can figure out what the input variable is and what the output variable is. In all of the examples above, the input variable is the variable t. This is because the differential equations that you’re most likely to see in biological models concern rates of change with respect to time. But this is not always the case, and the input variable might be location, the number of individuals in a population or anything else that might determine the value of another biologically relevant variable. For differential equations that depend on time, we can identify two basic types: pure time differential equations where the right hand side of the equation is purely a function of time, and does not depend on the output variable. In the list above, (a) is a pure time differential equation. Note that the right hand side might also be written in terms of a parameter. For example: where k is the name of the parameter that depends on the particular biological situation. We know that k is a parameter since simply writing
tells us that f is the output variable and t is the input variable. Since in this class we are only considering functions with one input and one output variable, k must be a parameter. The other basic type of equation is an autonomous differential equation. The name ‘autonomous’ comes from the stem autonomy. Autonomous differential equations are those that DO NOT depend on time. Example (b) above is an autonomous differential equation. The rate of change of the output variable b does not explicitly depend on time. That is, you can figure out the rate of change if you know the value of b, even if you don’t know what t is . The example (c) above is a mixed differential equation, since the rate of change depends both on the state variable f and on time t. Given any type of differential equation, you would like to come up with a solution to that differential equation. What does it mean to “solve” a differential equation? A solution to a differential equation is a function so that when you “plug that function into the equation” it becomes a true equation. For example, consider the pure time differential equation in (a) above: A solution to this differential equation is the function To check whether this is a solution, we simply take the derivative of f: so this solution checks out. You can check that the function is also a solution to this differential equation, so the solution is not unique. This is because if you add a constant, that constant goes away when you take the derivative. So we often talk about the general solution to the differential equation. In this case the general solution is any value). where c is an arbitrary constant (a constant that can take on To specify a particular solution to the differential equation, we need to specific the value that the function takes on at some point. For example, the problem might specify that and f(0)=12. To find the particular solution, we just find the general solution and plug in the value given: So is the particular solution to this differential equation. The need to specify an added condition along with an equation describing the change in a variable should seem familiar to you. In studying DTDS, the updating function specified how the state variable changed, but we needed to know the initial condition to determine the exact solution. In this case, the differential equation is like the updating function, and the v alue needed to find the particular solution plays the role of the initial condition.