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absolute value algebraic critical point analytic function analytic integral approaches zero arbitrary constant auxiliary equation becomes infinite change of variables characteristic curves characteristic equation circle complete integral condition congruence constant coefficients corresponding curve F defined degree denotes dependent determine different from zero double root elliptic function equa equal equations 64 essentially singular point example expression finite focal surface func given equation gral Hence identically independent variables infinite number infinitesimal transformation integral curve integral surface left.hand side Let us consider Let us suppose linear equation method movable singular multiplied neighborhood obtained origin partial derivatives partial differential equation particular integral point x0 poles polynomial preceding quadrature radius rational function reduces replace represented Riccati equation right.hand side single.valued function singular integral singular points straight line substitution system of integrals tangent plane theorem tion value y0 vanish yv y2
Page 232 - ... of the two parameters a and b from the three relations (53) and (54) will lead to the equation (52) and to that one only.* We shall now show that this equation (52) expresses the necessary and sufficient condition that the three equations (53) and (54) be satisfied by a system of three functions z, a, b of the two variables x and y, where p and q denote the partial derivatives of z with respect to x and y respectively. When this has been proved, it will be evident that the problem of integrating...
Page 197 - Q , ?/ 0 , ?/') = 0, for which this point is an ordinary point. is equal to the slope of the tangent to the curve (y) at this point. In this case we see first of all that the curve (y) is an integral curve of the equation (41). Moreover, it is an integral which is entirely unaccounted for in Cauchy's fundamental theorem, whatever may be the point chosen on the curve to fix the initial values of x and y. For if we take the point...
Page 277 - ... less than the number of independent variables. These are the two propositions which were to be established. Note 1. These propositions are of wide range : some special cases are worthy of special mention. Let...
Page 220 - Dm, and the plane tangent to any integral surface contains one of these straight lines. If we give the name characteristic curve to every curve which, at each of its points, is tangent to one of the corresponding m straight lines, the reasoning employed above shows again that every integral surface is a locus of characteristic curves. To obtain the differential equations of these curves, we are not compelled to carry out the decomposition of the left-hand side of the equation into linear factors....
Page 220 - R, we obtain equations of condition homogeneous in P, Q, R, which furnish m systems of values for the ratios of these coefficients for each point (x, y, z). Replacing P, Q, R in these conditions by the proportional quantities dx, dy, dz, we obtain the differential equations of the characteristic curves, and the integration of the partial differential equation is reduced to the integration of a system of ordinary differential equations. The preceding theory explains very simply how a linear equation...
Page 275 - ... of the integration of such an equation cannot, in general, be reduced to the integration of a system of ordinary differential equations. We can easily generalize, however, the method of the elimination of arbitrary functions which leads to a partial differential equation of the first order...
Page 275 - For when we differentiate the functional equation, first with respect to x and then with respect to y, we obtain the two equations tf'fcy) =/'(*). xf (X y) =/'(j,); and so, eliminating/'(a;i/), xf'(x) — yf'(y).
Page 192 - ... (x) is analytic except for poles, and let R be its radius. If the nearest singular point of X(x) to the origin were not a pole, we should take for C the circle through this singular point, and the function X (x) would then be analytic in this circle.
Page 91 - ... infinitesimal transformations has made it possible to apply the methods of the differential calculus to the theory of groups. Besides, in many questions concerning groups it is the infinitesimal transformation which is concerned, as we shall see from a few examples. Let us consider...