The nonlinear differential equation xx1x2 arises in the anal

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 + 2e^x

Implies d/dx(dy/dx) + x = 1 + 2e^x

Integrating both sides, we get

dy/dx + x²/2 = x + 2e^x +C₁

Integarting again, we get

y + x³/6 = x²/2 + 2e^x +C₁ x +C₂