280 Ordinary Differential Equations
Substituting i = dq/dt = q' into equation (6), we find that the charge on the
capacitor, q, satisfies the differential equation
(7) Lq" + Rq' + ~ = E(t).
Appropriate initial conditions are
(8) q(O) = qo and q' (0) = i(O) = io.
That is, appropriate initial conditions at some time t = 0 are the initial charge
on the capacitor, q 0 , and the initial current flowing in the system, io.
Differentiating equation (6) with respect to t and substituting i = dq/ dt =
q', we see the current, i, satisfies the second order linear differential equation
(9) Li"+ Ri' + .!:_ c = E' (t).
Appropriate initial conditions for this problem are to specify the initial cur-
rent, i 0 , and the initial rate of change of the current, i 0 , at some time t = 0.
That is, appropriate initial conditions are
(10) i(O) = io and i'(O) = i~.The initial current, i 0 , flowing in the circuit and the initial charge, qo, on
the capacitor are physically measurable quantities. But i 0 is not a physically
measurable quantity. However , if at t = 0 we measure io, qo, and Eo =
E(O), then we can calculate i 0 from equation (6). Substituting t = 0 into
equation (6) and solving for i~, we find
E R. qo
o - io - -
i ·'(0) - io ·t - L C
Comparing equation (7) with equations (3) and (5), we discover the cor-
respondence between electrical systems and mechanical systems (the simple
pendulum and spring-mass system) displayed in Table 6.2. This correspon-
dence allows us to simulate (model) mechanical systems, such as airplane
wings and suspension bridges- systems which would be expensive to actually
construct- by electrical systems- systems which are relatively inexpensive
to construct. By measuring quantities such as capacitance, inductance, resis-
tance, current, and voltage, we can determine the response of the approximat-
ing electrical system and can thereby infer the response of the hypothetical
mechanical system.