The nonlinear differential equation x”+x=1+x2 arises in the analysis of planetary motion using relativity theory. Classify (if possible) all critical points of the corresponding plane autonomous systems. Solution Here the Differential Equation is given by d2y/dx2 + x = 1 + 2ex Implies d/dx(dy/dx) + x = 1 + 2ex Integrating both sides, we get dy/dx + x2/2 = x + 2ex +C1 Integarting again, we get y + x3/6 = x2/2 + 2ex +C1 x +C2.