Applications: Some Model Problems Involving Differential Equations

Let us give here some typical examples of these problems that arise in applications:

Example 1: Mechanical Vibrations

Let denote the displacement at time of a block of mass m that is connected to a spring of stiffness k, a damper of resistance of an oscillator. If the system is released from position with a velocity, then its subsequent motion satisfies

This is an example of an initial value problem for a second order linear ODE.

Example 2: Ecology (Lotka-Volterra equations)

Consider a population of predators and prey living in an ecological niche. The predators survive by eating the prey and the prey exist on independent source of food. Let and , respectively, denote the population of predators and prey as time . Given the initial populations and of predators and prey, their subsequent populations satisfy the Lotka-Volterra equations:

where are positive constant corresponding to the prey's natural growth rate, the predator's natural death rate, the prey's death rate upon coming into contact with predators, and the predator's growth rate upon coming into contact with prey.

This example involves an IVP for a system of the first order non linear ODEs.