I think I've got an answer to this problem, but I am not 100% confident about this, so unless someone votes my solution up, do not accept it.
I believe if something is described by something else, the thing in question is all there is for the description to have been made.
On the other hand, if something is described with something else, the thing in question is a part of the solution, which helps the overall description.
To make the explanation easier, let's try some examples:
We describe the input with an exponential function.
In my opinion, this sentence means that the input was described by a longer list of functions, rules, definitions, which also contained an exponential function.
We describe the input by an exponential function.
This sentence means that the input was defined exactly and only by the exponential function, nothing more, nothing less.
To make it concrete, let's try a situation:
John, please, how many pills do I need to buy?
Answer A: 7 days multiplied by 3 pills a day = 21 pills.
Answer B: Well, that depends. If you're taking other pills, you can take only 2 per day, that would make it 7 days multiplied by 2 pills a day, 14 pills. It might also be possible we'll prolong our stay, in which case you'd need 14 days multiplied by 3 pills a day, that makes it 42 pills in total. You'd better get 42 pills to cover the worst case scenario.
In answer A, John resolved the problem by using an equation = the solution was described by the equation, just as the solution was described by John.
In answer B, John resolved the problem with a longer explanation, a part of which was an equation = the solution was described with the equation.
Again, I'm not 100% sure about this, it's still a long shot. I encourage other people to think about this and upvote/downvote this answer according to how they feel about it.