Much interest attaches to the determination of those linear differential equations which can be integrated by simpler functions, such as algebraic, elliptic, or Abelian. This has been studied by C. Jordan, P. Appel of Paris (born 1858), and
Poincaré.
The mode of integration above referred to, which makes known the properties of equations from the standpoint of the theory of functions, does not suffice in the application of differential equations to questions of mechanics. If we consider
the function as defining a plane curve, then the general form of the curve does not appear from the above mode of investigation. It is, however, often desirable to construct the curves defined by differential equations. Studies having this end in view have been carried on by Briot and Bouquet,
and by Poincaré.81
The subject of singular solutions of differential equations has been materially advanced since the time of Boole by G. Darboux
and Cayley. The papers prepared by these mathematicians
point out a difficulty as yet unsurmounted: whereas a singular solution, from the point of view of the integrated equation, ought to be a phenomenon of universal, or at least of general occurrence, it is, on the other hand, a very special and
exceptional phenomenon from the point of view of the differential equation.89 A geometrical theory of singular solutions resembling the one used by Cayley was previously employed by W. W. Johnson of Annapolis.
An advanced Treatise on Linear Differential Equations (1889) was brought out by Thomas Craig of the Johns Hopkins
University. He chose the algebraic method of presentation followed by Hermite and Poincaré, instead of the geometric
method preferred by Klein and Schwarz. A notable work, the
Traité d'Analyse, is now being published by Émile Picard of
Paris, the interest of which is made to centre in the subject of differential equations.