the forward Euler method
/ OY-ler /
Suppose you are standing on a hillside in fog with a compass that points straight downhill and a tape measure. You cannot see the valley, but you can read the slope right where your feet are. The simplest plan is: face the way the slope points, walk one fixed stride in that direction, then read the slope again at your new spot and repeat. Forward Euler is exactly this naive plan applied to a differential equation — follow the current slope blindly for one step, then re-measure.
The rule is y_{n+1} = y_n + h * f(t_n, y_n). In words: start at the current point (t_n, y_n), compute the slope f there, draw the straight tangent line with that slope, and slide along it for a horizontal distance h to land at the next point. Because the new value y_{n+1} appears only on the left and everything on the right is already known, the step is EXPLICIT — a single direct evaluation, no equation to solve. Geometrically you are approximating the true curved solution by a chain of short straight tangent segments, the way a polygon approximates a smooth arc. It is the discrete version of the definition of the derivative, (y(t+h) - y(t))/h is approximately f(t, y).
Forward Euler is the 'Hello, World' of ODE solvers: dead simple, cheap (one slope evaluation per step), and the foundation every fancier method builds on. But it is only FIRST order — the global error shrinks merely like h, so halving the step only halves the error, which is slow; getting three more correct digits needs a thousand times more steps. Worse, it is only conditionally stable: on a stiff or oscillatory problem an honest-looking step size can make the numerical solution explode while the true one decays. In practice it is a teaching tool and a building block, rarely the method you ship.
For y' = -2y, y(0) = 1 with h = 0.25: y_1 = 1 + 0.25*(-2*1) = 0.5, y_2 = 0.5 + 0.25*(-2*0.5) = 0.25, y_3 = 0.125. The true value at t = 0.75 is e^(-1.5) = 0.223, so Euler's 0.125 is noticeably low because each straight tangent overshoots the curve's bend. Try h = 0.6 instead: y_1 = 1 - 1.2 = -0.2, y_2 = -0.2 + 0.24 = +0.04 — it oscillates with the wrong sign, the first whiff of instability.
Follow the tangent for one step, then re-measure; too big a step on a decaying problem invites instability.
Do not confuse explicit (forward) Euler with implicit (backward) Euler: they share a name and an order but differ totally in stability. Forward Euler blows up on stiff problems unless h is tiny; backward Euler stays stable for any h.