An elementary treatise on differential equations and their applications . Fig. 3. lines through the origin. As the number of these is infinite, in this casean infinite number of characteristics pass through the singular poinj;. Ex.(ii). §*--?, *.^--l. ax y x ax This means that the radius vector and the tangent have gradients. Fia. 4. whose product is -1, that they are perpendicular. The char-acteristics are therefore circles of any radius with the origin as centre. * See a paper, Graphical Solution, by Prof. Takeo Wada, Memoirs of theCollege of Science, Kyoto Imperial University, Vol
An elementary treatise on differential equations and their applications . Fig. 3. lines through the origin. As the number of these is infinite, in this casean infinite number of characteristics pass through the singular poinj;. Ex.(ii). §*--?, *.^--l. ax y x ax This means that the radius vector and the tangent have gradients. Fia. 4. whose product is -1, that they are perpendicular. The char-acteristics are therefore circles of any radius with the origin as centre. * See a paper, Graphical Solution, by Prof. Takeo Wada, Memoirs of theCollege of Science, Kyoto Imperial University, Vol. II. No. 3, July 1917. GRAPHICAL REPRESENTATION In this case the singular point may be regarded as a circle of zero radius,the limiting form of the characteristics near it, but no characteristic offinite size passes through it. dy y — kx dx x + ky Writing dy/dx=ta,n\fs, y/x—tan 6, we get , tan 9 - ktan \/r = Ex. (iii). l+kt&nOtan \js + k tan \js tan 6=tan 6 - k,tan 0 — tan \fs _, -- -K, 1+tan Qt&n\]s tan {d-yf/) = k, a constant. The characteristics are therefore equiangular spirals, of whioh thesingular point (the origin) is the focus.
Size: 1593px × 1568px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1920, bookpublisherlondo, bookyear1920
