The existence theorem of ordinary differential equations

There are a great many devices for solving differential equations of certain special forms. But there is a large number of classes of differential equations that are not included in these special forms and cannot be integrated by quadratures or other purely elementary methods. When mathematicians were forced to abandon their cherished hope of finding a method for expressing the solution of every differential equation in terms of a finite number of known functions or their integrals they turned their attention to the question of whether a differential equation in general had a solution at all, and, if so, of what nature. This study resulted in the development of what is known as the Existence Theorem of Ordinary Differential Equations. This theorem states that for every ordinary differential equation of a fairly general type there exists a solution. The type of equations included in the theorem includes those that are usually encountered and used both in applied and pure mathematics. The theorem is no less important in the field of calculus than is the cardinal proposition in the theory or algebraic equations, that every such equation has a root