Chapter 3 - Applications of First-Order Ordinary Differential Equations

Abstract

This chapter provides an overview of applications of first-order ordinary differential equations. Two lines L1 and L2, with slopes m1 and m2, respectively, are perpendicular if the respective slopes satisfy the relationship m1 = -1m2. Hence, two curves C1 and C2 are orthogonal at a point if the respective tangent lines to the curves at that point are perpendicular. The chapter illustrates orthogonal trajectories and presents their solutions. It discusses population growth and decay assuming that the rate at which a population y(t) changes is proportional to the amount present. Mathematically, this statement is represented as the first-order initial value problem dydt = ky, y(0) = y0, where y0 is the initial population. If k > 0, then the population is increasing (growth), while the population decreases (decay) if k < 0. The chapter reviews Newton's law of cooling, which states that the rate at which the temperature T(t) changes in a cooling body is proportional to the difference between the temperature of the body and the constant temperature Ts of the surrounding medium. This situation is represented as the first-order initial value problem dTdt = k(T- Ts) , T(0) = To, where To is the initial temperature of the body and k is the constant of proportionality. The chapter illustrates some problems involving Newton's law of cooling in the examples presented in it.