Solution Curves of Equations in the Differential Space

Solutions of ordinary differential equations are considered. Differential Space is defined as the three-dimensional space with coordinates being the solution function and its first and second derivatives. Solution curves are represented as parametric three-dimensional curves in the differential space with the curve parameter being the independent variable. For various sample differential equations, the solution curves and their properties are depicted. The solution curves may converge to a point, may blow up and diverge to infinity, may be periodic, may end up with a limit cycle periodic solution or may be chaotic. Local solutions with a given initial condition set is treated in this introductory study. Differential space is a generalization of 2-D state space to 3-D’s. Global solutions with phase diagrams, basins of attraction are left for further studies.