An elementary treatise on differential equations and their applications . 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
An elementary treatise on differential equations and their applications . 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 Fig. 5. These three simple examples illustrate three typical a finite number of characteristics pass through a singularpoint, but an example of this would be too complicated to give-here.* * See Wadas paper. 10 DIFFERENTIAL EQUATIONS MISCELLANEOUS EXAMPLES ON CHAPTER I. Eliminate the arbitrary constants from the following : (1) y = Aex + Be~x + C. (2) y = Ae* + Be2x + C(?x. [To eliminate A, B, C from the four equations obtained by successive•differentiation a determinant may be used.] (3) y = ex(A cos x + B sin x), (4) y = c cosh-, (the catenary). Find the differential equation of (5) All parabolas whose axes are parallel to the axis of y. (6) All circles of radius a. (7) All circles that pass through the origin. (8) All circles (whatever their radii or positions in the plane xOy).[The result of Ex. 6 may be used. ] (9) Show that the results of eliminating a from 2y=x£+ax (1) and I from y = x^--bx2, (2) are in each case x2~-2x^- + 2y = 0 (3) [The complete primitive
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