5 - Differential equations

Abstract

The purpose of this chapter is to provide information on ordinary differential equations. It discusses how they can be used to model the behavior of systems in engineering and looks at their solution for different inputs to the systems. Differential equations arise from situations such as the lumped models designed to represent systems, the motion of projectiles, the cooling of a solid or liquid, transient currents and voltages in electrical circuits, oscillations with mechanical or electrical systems, and the rate of decay of radioactive substances. The term “solution” is used with a differential equation for the relationship between the dependent and independent variables such that the differential equation is satisfied for all values of the independent variable. The general solution consists of a family of equations that satisfy the differential equation; with a first-order differential equation, the general solution involves just one arbitrary constant. Given initial or boundary conditions, it is possible to find a value for the arbitrary constant and so obtain a particular solution. The order of a differential equation is equal to the order of the highest derivative that appears in the equation. The objective of this chapter is to teach how to represent engineering systems by differential equations, solve first- and second-order differential equations, and solve the differential equations representing models of engineering systems for step and ramp inputs.