Differential Equations

There is no more useful tool for the study of differential equations, in particular if they are in two dimensions, than the phase portrait. Many important systems both in physics and in economics in fact live in two dimensions. All second order systems are two dimensional. To this category belong all the oscillators, exemplified by the mathematical pendulum, or by the Samuelson-Hicks business cycle model if put in continuous time. It should be remembered that a second order differential equation, as characteristic of an oscillator, can always be put in the style of two coupled first order equations.