An elementary treatise on differential equations and their applications . e of afamily is defined as the locus of ultimate intersection of consecutive curves ofthe family. As thus defined it may include node- or cusp-loci in addition to orinstead of what we have called envelopes. (We shall give a geometrical reason forthis in Art. 56 ; see Lamb for an analytical proof.) tSee Lambs Infinitesimal Calculus, 2nd ed., Art. 155. If f(x, y, c) is ofthe form Lc^ + Mc + N, the result comes to M2=4LN. Thus, for y-cx-- = 0, i. e. c2x-cy + l = 0,the result is yl=ix. SINGULAR SOLUTIONS 67 Substituting in (
An elementary treatise on differential equations and their applications . e of afamily is defined as the locus of ultimate intersection of consecutive curves ofthe family. As thus defined it may include node- or cusp-loci in addition to orinstead of what we have called envelopes. (We shall give a geometrical reason forthis in Art. 56 ; see Lamb for an analytical proof.) tSee Lambs Infinitesimal Calculus, 2nd ed., Art. 155. If f(x, y, c) is ofthe form Lc^ + Mc + N, the result comes to M2=4LN. Thus, for y-cx-- = 0, i. e. c2x-cy + l = 0,the result is yl=ix. SINGULAR SOLUTIONS 67 Substituting in (1), y = ±2^x,or y2 = 4a\ This method is equivalent to finding the locus of intersection of f(x, y, c) =0,and f{x,y,c + h)=0, two curves of the family with parameters that differ by a smallquantity h, and proceeding to the limit when h approaches result is called the c-discriminant oif(x, y, c) = 0. 57. Now consider the diagrams 8, 9, 10, 8 shows the case where the curves of the family haveno special singularity. The locus of the ultimate intersections. FIG «. PQRSTUV is a curve which has two points in common with eachof the curves of the family ( Q and R lie on the locus and alsoon the curve marked 2). In the limit the locus PQRSTUV there-fore touches each curve of the family, and is what we have definedas the envelope. In Fig. 9 each curve of the family has a node. Two con-secutive curves intersect in three points ( curves 2 and 3 in thepoints P, Q, and R). The locus of such points consists of three distinct paris EE,AA, and BB. When we proceed to the limit, taking the consecutive curvesever closer and closer, AA and BB will move up to coincidencewith the node-locus NN, while EE will become an envelope. So 68 DIFFERENTIAL EQUATIONS in this case we expect the c-discriminant to contain the square ofthe equation of the node-locus, as well as the equation of the envelope. E-
Size: 1844px × 1356px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1920, bookpublisherlondo, bookyear1920
