SINGULAR SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS
SINGULAR SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS
复制标题
一阶微分方程的奇异解
DOI：
10.1093/qmath/os-3.1.238
复制
发表时间：
1932
期刊：
影响因子：
--
通讯作者：
T. Chaundy
中科院分区：
文献类型：
--
作者：
T. Chaundy
微分方程4>(x, y, p)= 0,(1)的奇异解，无论是由通解j (x, y, c)= 0的c-判别式求得，还是由< f>(x, y, p)= 0本身的^-判别式求得，我们知道，可能伴随着分别对应于节点轨迹或节点轨迹的外来因子。即使一个因子同时出现在两个判别式中，也不一定是奇异解，因为一个斜轨迹同时被认为是一个节点轨迹和一个顺轨迹。因此，有必要进行进一步的测试。然而，如果我们考虑在某些标准解方法中自然出现的参数形式的通解x= x (p, c), y= y (p, c),(2)，我们可以确定不受无关因素影响的奇异解。奇异解作为由通解组成的曲线族的包络，将在一个（或多个）点与该族的每条曲线接触，并且这些点的集合充分构成包络。那么，假设包络线在参数化给出的点处接触曲线c=常数
THE singular solution of the differential equation4>(x, y, p)= 0,(1) as obtained either from the c-discriminant of the general solution j (x, y, c)= 0 or from the^-discriminant of< f>(x, y, p)= 0 itself, may, we know, be accompanied by extraneous factors corresponding respectively to a nodal locus or a tac-locus. Even a factor which appears in both discriminants is not necessarily a singular solution, since a cuspidal locus reckons both as a nodal locus and as a taclocus. Further tests are therefore necessary.* If, however, we consider the general solution in the parametric form x= x (p, c), y= y (p, c),(2) which occurs naturally in certain standard methods of solution, we can determine the singular solution unencumbered with irrelevant factors, fThe singular solution, being the envelope of the family of curves comprised by the general solution, will touch each curve of the family at a point (or points), and the aggregate of these points sufficiently constitutes the envelope. Suppose, then, that the envelope touches a curve c= constant at the point given parametrically by