Were integrals really called solution curves (or vice versa)?

Question

For some reason I recall hearing that around the time Euler wrote his Calculus books (1768-1770), or even before then, what we call integrals now were called solution cuvres (or even possibly the other way around since my memory on this is seriously hazy). Can I get some confirmation or refutation on this?

Answer 1

They are still so called in some contexts, although "integral curves" is more common. If $dy/dx=X(x)$ then $y$ is both an integral of $X$ and a solution to the differential equation. Such dual use does already appear in Euler's Institutionum Calculi Integralis. For example, he writes in §11 (in Bruce's translation):

"Now even if here only a single variable quantity $x$ appears, yet in fact two are considered; for the other is that function itself, the differential of which we take to be $Xdx$; which if we designate with the letter $y$, becomes $dy=Xdx$ or $dy/dx=X$, thus as here generally is the ratio of the differentials $dy:dx$ proposed, which is equal to $X$, and thus there becomes $y=\int Xdx$."

And later in §§36-39 he talks about what is now called general and particular solutions to a differential equations, but in the context of (ordinary) integrals:

"Hence the complete integral embraces all the particular integrals within itself and from that all these are able to be formed easily. But in turn from the particular integrals the general integral cannot become known. But many times, as henceforth it becomes apparent, a method of finding the complete integral from the particular integral is obtained."

He continues this dual use when he gets to solving the differential equations proper, e.g. in §§411-415 the problem of solving various examples is stated as:"To find the integral for this proposed differential equation".