Differential Equations

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Submitted By tarungehlot
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Differential Equations by tarungehlots

Concepts of Differential Equation Consider a variable that might denote the per capita capital stock level in an economy. When convenient, we can recognize that the level of capital depends upon time by addition the time argument. Doing so, we would write the per capita capital stock level as , rather than just writing . If we think of time as unfolding continuously, we can also think of capital as being a continuous function of time, and we can assume that the function has derivate . This derivative is the instantaneous change in per capita capital. In many presentations, is written as , but we will here use the more familiar notation . By writing the derivate as , so that we include the time argument, we are emphasizing that the value of the derivative may change with time. Because it is typically cumbersome to repeatedly write down the argument, we can also write the derivative as just , while remembering that its value may change with time. A differential equation is an equation that relates the time derivative of a variable to its level. An example is the equation (1) . The variable is called a state variable because it gives the state of the system at any given point in time. In our example, gives the state or level of per capita capital stock.

2. A Dynamic System

The basic dynamic principle is the idea that “the way things are determines the way things change.” A differential equation is one way of modeling the basic dynamic principle. The state variable indicates the way things are, while the time derivative indicates how things change. The differential equation itself is what relates the two. More precisely, a differential equation typically presents a functional relationship, showing how depends upon . When we want to…...

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