Step-by-Step Approximation

Calculus 2 · Axiom Academy

Most differential equations have no exact formula for their solution. So you do the next best thing: follow the slope a little at a time. That is Euler's method. When you can't solve it, you can still trace it A differential equation hands you something modest at every point: a slope . It tells you which way the solution heads from wherever you are — but not, in general, a tidy formula for the curve itself. The trick of Euler's method is to take that slope at face value: stand at the start, head in the direction the equation points, walk a short step, then ask again. Here the equation is the one below: at any point (x,y) the solution's slope is x+y , and we start from y(0)=1 . The little dashes are that slope drawn everywhere — a slope field . Press play and watch the path get built one straight step at a time, each step aimed along the dash beneath it, landing on a connect-the-dots approximation of the true solution. Eight straight steps already trace the shape — though the corners hug the inside of the bend, always a little below the true curve. Eight steps got close. What if you took more? Drag the slider to change how many steps N you split [0,1] into — the step size h shrinks, the corners pile up tighter against the true curve, and the gap at the finish line collapses. Watch h and the endpoint error fall together. Halve the step and you roughly halve the error — the price of that accuracy is twice the steps to compute. Each step just follows the slope the equation gives

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