Autonomous Equations: Slope Field and Numerical Solutions

This GeoGebra example visualizes autonomous differential equations by displaying their slope fields (direction fields) and numerical solution curves. The core mathematical concepts include autonomous ODEs (where the right-hand side does not explicitly depend on the independent variable), slope fields, initial value problems, and numerical integration methods such as Euler or Runge-Kutta. The construction uses GeoGebra commands: SlopeField to generate the direction field, SolveODE to compute numerical solutions, Length, Vector, and Segment to build direction and normal vectors, Textfield and InputBox1 for interactive input of the equation or initial conditions, h as the step size, B as a draggable point, and numericalIntegral1 and numericalIntegral2 to display integral curves for different initial values. This resource is ideal for university-level calculus or differential equations courses, helping students understand equilibrium points, stability, and phase-line analysis, and can also serve as an advanced enrichment for high school mathematics.