Chapter 12 - Differential Equations

Abstract

The solution of a differential equation is a function whose derivative or derivatives satisfy the equation. An equation of motion is a differential equation obtained from Newton's second law of motion, F=ma. In principle, an equation of motion can be solved to give the position and velocity as a function of time for every particle in a system governed by Newton's laws of motion. Some differential equations can be solved by separation of variables. A homogeneous linear differential equation with constant coefficients can be solved by use of an exponential trial solution. An inhomogeneous linear differential equation can be solved if a particular solution can be found. An exact differential equation can be solved by a line integration. Some inexact differential equations can be converted to exact differential equations by multiplication by an integrating factor. Some partial differential equations can be solved by separation of variables. Some differential equations can be transformed into algebraic equation by a Laplace transformation. Solution of this equation followed by inverse transformation provides a solution to the differential equation. Differential equations can be solved numerically by a variety of methods, including the use of Mathematica.