Mathematics & Science
Applications of Differential Equations
(continued from last page...)
A Preliminary Example
First let's consider a circuit with the following values:
We'll remove the voltage source from our circuit, i.e. we'll let E(t) = 0. (Such a circuit is said to have free electrical vibrations.)
Of course, since we're working with a second order differential equation we also need two initial conditions. In this case let them be:
(Note: these initial conditions imply that the initial current is zero, and that the capacitor is holding an initial charge of 1 coulomb. Since E(t) = 0, the only reason any current at all starts moving through the circuit is this initial capacitor charge.)
Shortly we'll have Mathematica solve the initial value problem, but before we do, take another look at the initial conditions. They refer to two different functions, I and q.
Since we're looking for current as a function of time we'll be solving equation (3) from the Introduction:
L I'' + R I' + I/C = E',
so to solve it uniquely, we really need to know both I(0) and I'(0). Obviously I(0) we have already, but how can we get I'(0) from knowing q(0)?
Well let's look at equation (1) from the introduction again:
LI' + RI + q/C = E
It relates I and q, (both of which we know at t=0), to I'. All we need to do is solve this equation for I'. Doing this we get:
I' = (E - RI - q/C) / L
Now letting t=0, we get an expression for I'(0):
I'(0) = (E(0) - R I(0) - q(0)/C) / L
Every one of the values on the right-hand side of this equation is already known. Look higher up this page, and you'll see them. Actually, since this is our first example, I'll summarize them for you:
In the next problem you'll be on your own in gleaning this information from the problem's description.
Anyway, time for you to do some work. Substitute the above list of known values into the formula for I'(0). Launch Mathematica and use it to calculate the actual decimal value of the expression. (You're using Mathematica as a simple calculator here.)
ODE Laboratories: A Sabbatical Project by Christopher A. Barker
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