Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 6.2: Illustration of the one-parameter family of solutions to y ′ =
(y − C)/τ. As t → ∞, all solutions tend towards C.
C
If y 0 − C ≠ 0 then the second term will be nonzero. The exponential
factor is always decreasing, so the second term decreases in magnitude with
time. Consequently the value of y → C as t → ∞. To emphasize this we
may chose to replace the symbol C with y ∞ (this is common in biological
modeling) giving
y = y ∞ + (y 0 − y ∞ )e −t/τ (6.24)
Example 6.2. RC-Circuits. An example from engineering is given by
RC-circuits (see figure 6.3). The RC refers to a circuit that contains a
battery (or other power source); a resistor of resistance R and a capacitor
of capacity C. If there is an initial charge q on the capacity, then the charge
will decay exponentially if the voltage is sent to zero. RC circuits have the
following rules Electric Circuits are governed by the following rules:
1. The voltage drop across a resistor (the difference in voltages between
the two ends of the resistor) of resistance R with a current i flowing
through it is given by ∆V resistor = iR (Ohm’s Law)
2. Current i represents a flow of charge i = dq/dt.
3. If there is a voltage drop across a capacitor of capacitance C, there
will be a charge q = CV on the capacitor, hence (by differentiating),
there is a current of i = CdV/dt.
4. The voltage drop across an inductor of inductance L is ∆V inductor =
Ldi/dt.
5. The total voltage drop around a loop must sum to zero.