Approximating Solutions Using Euler's Method

A differential equation like dy/dx = x + y tells you the slope at every point, but not the curve itself. Euler's method builds the curve one short straight step at a time. Start at the known point. Use the differential equation to find the slope there. Walk a short horizontal distance h along that tangent line to get a new y value. That new point is only an estimate, not the true curve, but it is close if h is small. Then repeat: find the slope at the new point, step again. Each step uses whatever point you just landed on, even though that point is already slightly off the real curve.

Mechanically, each step is y_new = y_old + h times (slope at x_old, y_old), and x_new = x_old + h. You recompute the slope fresh at every step because the slope formula depends on x and y, and both have changed. Writing the steps out in a small table of x, y, and slope keeps the arithmetic organized and is exactly what a worked problem expects.

This topic sits right after slope fields, because Euler's method is really a slope field walk written as arithmetic instead of a picture. It matters most when a differential equation is separable in theory but ugly or impossible to solve by hand, or when you just want a numerical estimate at one specific x value. Smaller steps track the true curve more closely, but the method never becomes exact at any positive step size, since it always follows a straight segment instead of the actual curving solution.