A differential equation is an equation that involves one or more derivatives. Many physical problems, when formulated mathematically, lead to differential equations. For example, the equation (k > 0)
describing the decrease in y as a function of time, occurs in the fields of reaction kinetics, radiochemistry or electrical engineering (where y represents concentration of a chemical species, or atoms of a radioactive element, or electrical charge, respectively) as well as in many other fields. Of course, a differential equation can be more complicated that the one shown in equation 101; another example from electrical engineering is shown in equation 10-2, where R is the resistance in a circuit, L is the inductance, E is the applied potential, i is the current and t is time.
If a differential equation contains derivatives of a single independent variable, it is termed an ordinary differential equation (ODE), while an equation containing derivatives of more than one independent variable is called a partial differential equation (PDE). Partial differential equations are discussed in a subsequent chapter.
The general form of an ordinary differential equation is
and although writing the differential equation, such as the above, may be simple, solving the problem is not. By "solving," we mean that we want to be able to calculate the value of y for any value of x. Some differential equations, such as 10-1, are solvable by symbolic integration (the integrated equation is Iny = -kt + const), but many others may not be amenable to solution by the "pencil-and-paper" approach. Numerical methods, however, can always be employed to find the value of the function at various values of t. Although we haven't found an expression for the function F(x, y), but simply obtained a table of y values as a function ofx, the process is often referred to as "integration."
You may remember from your freshman calculus class that when an expression is integrated, an arbitrary constant of integration is always part of the solution. For example, when equation 10-1 is integrated, the result is lny = ~kt + In y0, or y, = yoe"k'. A similar situation pertains when numerical methods are employed: to solve the problem, one or more values of the dependent variable and/or its derivative must be known at specific values of the independent variable. If these are given at the zero value of the independent variable, the problem is said to be an initial-value problem; if they are given at some other values of the independent variable, the problem is a boundary-value problem. This chapter deals with initial-value problems, while the following chapter deals with boundary-value problems.
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