Numerical Integration of Ordinary Di„erential Equations

For initial value problems, we examine several numerical integration methods, including the Euler method, the modi›ed Euler method, the Runge-Kutta method, and the built-in ode45 function of MATLAB. For boundary value problems, we employ implicit ›nite-dižerence methods that reduce ODEs to a system of linear algebraic equations, which turn out to be a tri-diagonal system that can be solved algebraically.