1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 281

Table 6.2 Correspondence between electrical and mechanical systems

Electrical systems Mechanical systems

L inductance m m ass

q charge y displacement

dq/dt = i current dy/dt = v velocity

R resistance c damping

1 /C reciprocal of k spring or

capacitance pendulum constant

E(t) electromotive force f(t) driving force

6 .1.1 Free Motion

Since the simple p endulum, mass on a spring, and the RLC series circuit

all lead to the initial value problem

(11) ay" +by'+ dy = f(t); y(O) =co, y'(O) = c1

where a > 0, b ~ 0 , and d > 0 are constants, we should examine this initial

value problem in detail.

When there is no external force driving the system- that is, when f(t) = 0
for all t the system is said to execute free motion. In this case the differential
equation of (11) reduces to the homogeneous different ial equation

(12) ay" + by' + dy = 0

and the associated a uxiliary equation is ar^2 + br + d = 0. The two roots of

the auxiliary equation are

(13)

-b + ,/b^2 - 4ad

2a

and

- b-Vb^2 - 4ad

r2=----- -

2a

(NOTE: These roots can easily be calculated using a root finding routine

instead of using the quadratic formula (13).)