Linear differential equations with constant coefficients are very common in physical and chemical science, and of these, the simplest and most frequently met is the first-order equation a
dy
/
dt
+
y
=
f(t)
, (1) where
a
is a constant, and
f(t)
a single-valued function of
t
. The equation signifies that the quantity
y
is removed at a rate proportional to the amount present at each instant, and is simultaneously restored at a rate dependent only upon the instant in question. Familiar examples of this equation are the charging of a condenser, the course of a monomolecular reaction, the movement of a light body in a viscous medium, etc. The solution of this equation is easily shown to be
y
=
e
-
t
/
a
{
y
0
= 1 / a ∫
t
0
e
t
/a
f(t)
dt
, (2) where
y
0
is the initial value of
y
. In the case where
f(t)
= 0, this reduces to the well-known exponential decay of
y
.