Any equation involving the derivatives, y¢, y², … , of a function y is called a differential equation (DE). Any function that satisfies the DE is called a solution. The order of a DE is the order of the highest derivative in it. Examples:
(a) is 1st order with solutions , where c is any constant.
(b) is 2nd order with solutions , where c1 and c2 are arbitrary constants.
Experiences have shown that it is hopeless to seek methods for solving all DE’s. Instead, it is more fruitful to ask whether a given DE has any solution, and what properties of the solutions, if exist, can be deduced from the DE. Thus, DEs provide a new source of functions.
Results of great generality about the solutions of DEs are usually difficult to obtain. However, there are a few exceptions, e.g., the linear DEs that are our only concern.