PlanetPhysics/Differential Equations

General Remarks.


A differential equation is any equation involving derivatives of a dependent variable with respect to one or more independent variables. Thus





are classified as differential equations. Equation (3) is called a partial differential equation for obvious reasons. The others are called ordinary differential equations. One may have a system of differential equations involving more than one dependent variable,



In addition to their own mathematical interest differential equations are particularly important since the scientist attempts to describe the behavior of certain aspects of the Universe in terms of differential equations. We list a few of this type.






The order of the highest derivative occurring in a differential equation is called the order of the differential equation.

Solution of a Differential Equation. Initial and Boundary Conditions


Let a differential equation be given involving the dependent variable   and the independent variables   and  . Any function   which satisfies the differential equation is called a solution of the differential equation. For example, it is easy to prove that   satisfies


We say that   is a solution of  . It is important to realize that a differential equation has, in general, infinitely many solutions. For example,   admits any function   as a solution, where   and   are constants which can be chosen arbitrarily. To specify a particular solution, either initial conditions like


or boundary conditions like


must be given in addition to the differential equation. In the first case  is the solution, and in the second case   is the solution. A full study of a differential equation implies the determination of the most general solution of the equation (involving arbitrary elements which may or may not be constants) and a discussion of how many additional conditions must be imposed in order to fix uniquely the arbitrary elements entering in the general solution. Without attempting an exact statement or proof, at the moment, we state the fact that "in general" the most general solution of an ordinary differential equation of order   contains exactly   arbitrary constants to be uniquely determined by   initial conditions.



[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.

This entry is a derivative of the Public domain work [1].