Evaluate the expression when x = 8 and y = 1.
2 answers:
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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:
When two quantities
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
Now, when
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
This is a separable ODE: