## Integration of Ordinary Differential Equations |

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arbitrary constant Bernoulli equation complementary function condition constant coefficients curve of contact cusp dependent variable derivatives differentiating with respect discriminant locus eliminating envelope equation becomes equation F(x equation of order equivalent exact Example exponential expression family of curves family of integral finite gradient Hence homogeneous type horizontal plane identically imaginary indicial equation integral curves integrating factor intersections inverse operator involving left-hand member linear equation linearly independent obtain ordinary differential equation origin orthogonal trajectories p-discriminant parabolas parameter particular integral particular solution plane curves polar co-ordinates polynomial polynomial of degree positive integer primitive primitive equation radius of curvature reduced equation replaced represents a family satisfied second degree second order Separating the variables singular integral singular point singular solution straight lines substitution surface tac-locus tangent transformation variables and integrating variables are separable whence write written z-axis zero