Solution of a Differential Equation

A solution of the ordinary differential equation (11) on the interval is a function such that exist and satisfy

for every t in . Unless stated otherwise, we assume that the function f of Eq. (11) is a real-valued function, and we are interested in obtaining real-valued solutions .

It is easily verified by direct substitution that the first order equation (3),

has the solution

where c is an arbitrary constant.

Although for the equation (3), we are able to verify that certain simple functions are solutions, in general we do not have such solutions readily available. Thus a fundamental question is the following: Does an equation of the form (11) always have a solution? The answer is "NO." Merely writing down an equation of the form (11) does not necessarily mean that there is a function that satisfies it. So, how can we tell whether some particular equation has a solution? This is the question of existence of a solution, and it is not a purely mathematical concern, for at least two reasons.

1. If a problem has no solution, we would prefer to know that fact before investing time and effort in vain attempt to solve the problem. If a sensible physical problem is expressed mathematically as a differential equation, then the equation should have a solution. If it does not, then presumably there is something wrong with the formulation. In this sense an engineer or scientist has some check on the validity of the mathematical problem.

2. Assuming that a given differential equation has a least one solution, the question arises as to how many solutions it has, and what additional conditions must be specified to single out a particular solution. This is the question of uniqueness. In general, solutions of differential equations contain one or more arbitrary constants of integration, as does the solution (13) of Eq. (3). Equation (13) represents an infinity of functions correcsponding to the infinity of possible choices of the constant c. If R is specified at some time t, this condition will determine a value for c---even so, we do not yet know whether there are other solutions of Eq. (3) that also have the prescribed value of R at the prescribed time t. The questions of existence and uniquesness are difficult questions

A more practical concern is: given a differential equation of the form (11), can we actually detemine a solution, and if so, how? Note that if we find a solution of the given equation, we have at the same time answered the question of the existence of a solution. However, without knowledge of existence theory we might, for example, use a computer to find a numerical approximation to a "solution" that does not exist. On the other hand, even though we may know that a solution exists, it may be that the solution is not expressible in terms of the usual elementary functions---polynomial, trigonometric, exponential, logarithmic, and hyperbolic functions. Unfortunately, this is the situation for most differential equations arising in scientific applications.