Brief review of proportionality relationships:
When two quantities
are
proportional, that means any change in
manifests a
change (think "in the same direction") in
.
Silly example: "The more I eat, the fatter I get." Here the amount one eats is directly proportional to one's body weight.
This change isn't always one-for-one, so we introduce a constant
to account for any scaling that occurs on either variables behalf. In general, though, we can write a directly proportional relationship as
.
Now, when
are
proportional, then a change in
manifests a change in
in the
(opposite) direction.
Silly example: "The more I eat, the less thin I get."
This time we write the relation as
.
To get back to your problem: To say that the rate of change of
is inversely proportional to
is to say that there is some constant
such that
This is a separable ODE:
