Audio, Video, and Tabbed Content

Euler’s method in Python:

def euler(f, y0, t_span, dt):
    """Solve dy/dt = f(t, y) with initial value y0."""
    t_start, t_end = t_span
    t, y = t_start, y0
    result = [(t, y)]
    while t < t_end:
        y = y + dt * f(t, y)
        t = t + dt
        result.append((t, y))
    return result

# Example: dy/dt = -y, y(0) = 1 → exact: y = e^{-t}
solution = euler(lambda t, y: -y, 1.0, (0, 5), dt=0.01)

Euler’s method in TypeScript:

function euler(
  f: (t: number, y: number) => number,
  y0: number,
  tSpan: [number, number],
  dt: number,
): [number, number][] {
  const [tStart, tEnd] = tSpan;
  let t = tStart, y = y0;
  const result: [number, number][] = [[t, y]];
  while (t < tEnd) {
    y = y + dt * f(t, y);
    t = t + dt;
    result.push([t, y]);
  }
  return result;
}

// Example: dy/dt = -y, y(0) = 1
const solution = euler((t, y) => -y, 1.0, [0, 5], 0.01);

Euler’s method in Rust:

fn euler<F>(f: F, y0: f64, t_start: f64, t_end: f64, dt: f64) -> Vec<(f64, f64)>
where
    F: Fn(f64, f64) -> f64,
{
    let mut t = t_start;
    let mut y = y0;
    let mut result = vec![(t, y)];
    while t < t_end {
        y += dt * f(t, y);
        t += dt;
        result.push((t, y));
    }
    result
}

// Example: dy/dt = -y, y(0) = 1
fn main() {
    let solution = euler(|_t, y| -y, 1.0, 0.0, 5.0, 0.01);
    println!("y(5) ≈ {:.6}", solution.last().unwrap().1);
}