Page 209 - Foundations Of Differential Calculus

192 9. On Differential Equations
Hence unless this finite equation is satisfied, the given equation will be
imaginary.
327. We might have treated in this chapter differential equations of higher
order that contain three variables, and we might have considered and de-
cided which of these turn out to be either real or imaginary. However, since
the criteria become extremely intricate, we omit this work, especially since
this follows from the same sources which we have here explored. Indeed, if
there is need for these criteria in integral calculus, at that stage they can
easily be developed. For the same reason we have not at this time consid-
ered equations with more variables, especially since they practically never
occur. If it is ever necessary, there should be no difficulty in examining such
equations with the principles we have discussed here. For these reasons we
here bring to a conclusion our exposition of the principles of differential cal-
culus. We next move on to show some of the more important applications
that this calculus has both in analysis and in higher geometry.