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arbitrary constant Bessel Bessel equation C. A. Coulson complementary function constant coefficients cos2 cosh curve of contact dependent variable differentiating with respect discriminant locus distinct eliminating envelope equation F(x equation of order equivalent Example expression family of curves finite gradient Hence homogeneous type horizontal plane indicial equation integral curves integrating factor intersections involving left-hand member Legendre equation linear equation linearly independent obtain ordinary differential equation origin orthogonal trajectories p-discriminant parabolas parameter particular integral particular solution plane curves Pn(x polynomial polynomial of degree positive integer primitive quadratures radius of curvature reduced equation replace represents a family roots satisfied second degree second order second solution Separating the variables singular integral singular point singular solution straight lines substitution subtangent surface tac-locus tangent transformation variables and integrating variables are separable whence write written zero