Should I use by construction or by assumption?

Question

I'm solving a differential equation with some boundary condition. When I use the boundary condition in my solution, should I say ... is true by construction or by assumption? Or should I use some other expression?

Answer 1

Generally speaking, I would use "by construction" when you are "building" something in order to satisfy peculiar properties you are interest in so that you can use it in your proof and/or when someone else has done it before and you are just talking about his structure. If there is some given definition (that you have set by yourself at the beginning of your research paper, maybe) that you are referring to, you can simply use "by definition (see definition number X.YZ)" in order to invoke it. Now, in your example, you are not assuming anything, just taking the boundary condition (set of contraints, initial condition/data or whethever you wish) in order to solve a calculus problem... so, IMHO, it would be sufficient to call it with its proper name (i.e., just say "by the boundary condition we have set at the beginning" or something like that).