TY - JOUR
AU - Izumiya, S.
AB - SINGULAR SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS S. IZUMIYA Dedicated to the memory of Professor Giko Ikegami 0. Introduction In classical treatises of equations (Caratheodory [2], Courant and Hilbert [3], Forsyth [4, 5], Ince [8], Petrovski [14]) the discussions of equations with singular solutions are informal. In these, definitions of singular solutions are very confused. Even in modern articles [9, 12], they are studied under special assumptions. In this note we shall give a rigorous definition of singular solutions of first-order differential equations for real-valued functions (Theorem A). On the other hand, complete integrability is an important notion for the classical theory of first-order differential equations. The notion of singular solutions has usually appeared in the above articles accompanied by the notion of complete solutions. Recently, we have studied some generic properties of completely integrable systems of first-order differential equations as an application of the theory of Legendrian unfoldings [7,10,11]. However, we have never seen a characterization of complete integrability. Our other purpose is to give a characterization of complete integrability of first-order differential equations (Theorem B). In Section 1, we shall state our main results. The proof of Theorem A will be given in Section 2. We shall prove 
TI - Singular Solutions of First‐Order Differential Equations
JF - Bulletin of the London Mathematical Society
DO - 10.1112/blms/26.1.69
DA - 1994-01-01
UR - https://www.deepdyve.com/lp/wiley/singular-solutions-of-first-order-differential-equations-2n7E80k00h
SP - 69
EP - 74
VL - 26
IS - 1
DP - DeepDyve
ER -