Differential Equations

Abstract

In this chapter we deal with differential equations. After defining differential equations we proceed with first order linear differential equations but we also discuss some nonlinear important examples: the Bernoulli and the Riccati equations. The latter is used to investigate the saturation of markets, the logistic growth. As linear differential equations of second order are very important in mathematical modelling they are discussed in full detail. At last we deal with a ubiquitous model in economics and biology the predator-prey model and the Lotka-Volterra differential equations, whose solutions are based on the Poincaré-Bendixson theorem.

A (system of) differential equation(s) relates some unknownfunction(s) with some of its (their) derivatives. In applications,the functions usually represent physical, engineering,biological or economic quantities, and the derivatives representtheir rates of change.