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.