So I have two equations, and three unknowns.... Have I lost information somewhere? Is it possible to eliminate all unknown constants?? I can't see how, but the question seems to imply that this is what is expected.....

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C can be any number you want to be.

Here is my logic:

x=...+C

dx/dt=... without C

dx/dt will be fine for any set of functions x=... +C

I am not sure about this. It is just an idea