Integration of Ordinary Differential Equations

The solution of differential equations is an important problem that arises in a host of areas. Many differential equations are too difficult to solve in closed form. Instead, it becomes necessary to employ numerical techniques. Differential equations have a major application in understanding physical systems that involve aerodynamics, fluid dynamics, thermodynamics, heat diffusion, mechanical oscillations, etc. They are used for developing control algorithms for dynamic simulations. Other applications also include optimization and stochastics. We will consider ordinary differential equations (ODEs) and focus on two classes of problems: (a) first-order initial-value problems and (b) linear higher-order boundary-value problems. The basic principles we will see in these two classes of problems carry over to more general problems. For instance, higher-order initial-value problems can be rewritten in vector form to yield a set of simultaneous first-order equations. First-order techniques may then be used to solve this system. Consider a general nth order differential equation in the form