Mathematics - Numerical Methods - Numerical solutions to ODEs - ODE Numerical Methods

10 steps: (1) PROBLEM: solve dy/dx=f(x,y), y(x₀)=y₀ numerically. (2) EULER'S METHOD: y_{n+1} = yₙ + h·f(xₙ,yₙ). Follow the slope. (3) Error: O(h) per step, O(h) globally. First-order method. (4) IMPROVED EULER (Heun's): predict with Euler, then average slopes. (5) y_{n+1} = yₙ + h/2[f(xₙ,yₙ)+f(x_{n+