An Elementary Treatise on Differential Equations and Their Applications
by H.T.H. Piaggio
"The Theory of Differential Equations, said Sophus Lie, "is the most important branch of modern mathematics." The subject may be considered to occupy a central position from which different lines of development extend in many directions. If we travel along the purely analytical path, we are soon led to discuss Infinite Series, Existence Theorems and the Theory of Functions. Another leads us to the Differential Geometry of Curves and Surfaces. Between the two lies the path first discovered by Lie, leading to continuous groups of transformation and their geometrical interpretation. Diverging in another direction, we are led to the study of mechanical and electrical vibrations of all kinds and the important phenomenon of resonance. Certain partial differential equations form the starting point for the study of the conduction of heat, the transmission of electric waves, and many other branches of physics. Physical Chemistry, with its law of mass-action, is largely concerned with certain differential equations. The object of this book is to give an account of the central parts of the subject in as simple a form as possible, suitable for those with no previous knowledge of it, and yet at the same time to point out the different directions in which it may be developed. The greater part of the text and the examples in the body of it will be found very easy. The only previous knowledge assumed is that of the elements of the differential and integral calculus and a little coordinate geometry. The miscellaneous examples at the end of the various chapters are slightly harder. They contain several theorems of minor importance, with hints that should be sufficient to enable the student to solve them. They also contain geometrical and physical applications, but great care has been taken to state the questions in such a way that no knowledge of physics is required. |