Euler’s Method

For \(y^{\prime}=\mathrm{f}(x,y)\), start from \((x_0,y_0)\) and use the current point’s slope for a straight step of constant length \(h\).

\(x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h\mathrm{f}(x_n,y_n)\).

Example

For \(y^{\prime}=x+y\), \(y(0)=1\) and \(h=0.1\):

Thus \(y(0.2)\approx1.22\). Keep unrounded intermediate values. This is an approximation; a smaller step often improves accuracy over a fixed interval, but is not a guarantee for every problem.