Chapter 11 - Applications of Systems of Ordinary Differential Equations

Abstract

Electrical circuits can be modeled with ordinary differential equations. This chapter illustrates how a circuit involving loops can be described as a system of linear ordinary differential equations with constant coefficients. This derivation is based on these principles: (1) Kirchhoff's current law and Kirchhoff's voltage law. In determining the drops in voltage around the circuit, the voltages are added consistently in the clockwise direction. The positive direction is directed from the negative symbol toward the positive symbol associated with the voltage source. In summing the voltage drops encountered in the circuit, a drop across a component is added to the sum if the positive direction through the component agrees with the clockwise direction. Otherwise, this drop is subtracted. In the case of the L-R-C circuit with one loop involving each type of component, the current is equal around the circuit by Kirchhoff's Current Law. Derivation of the modeling differential equation becomes more complicated as the number of loops in the circuit is increased. In this case, the current through the capacitor is equivalent to i1–i2. The chapter also discusses the circuit made up of three loops. It analyzes the problems involving the determination of the diffusion of a material in a medium lead to first-order systems of linear ordinary differential equations. The motion of a mass attached to the end of a spring can be modeled with a second-order linear differential equation with constant coefficients. This situation can be expressed as a system of first-order ordinary differential equations as well. Population problems based on the simple principle that the rate at which a population grows or decays is proportional to the number present in the population at any time t can be extended to other examples involving more than one population that lead to systems of ordinary differential equations.