Euler's Method

Numerically approximating a solution curve by taking small straight-line steps, $y_{n+1}=y_n+h\cdot f(x_n,y_n)$ .

Euler's Method approximates a differential equation's solution numerically, without solving it symbolically: starting from a known point $(x_0,y_0)$ , take small steps of size $h$ , at each step moving along the slope the differential equation gives at the CURRENT point: $y_{n+1}=y_n+h\cdot f(x_n,y_n)$ . Small steps (small $h$ ) track the true solution more closely, at the cost of more steps to cover the same distance.

Examples

For $y'=x+y$ , $y(0)=1$ , with step $h=0.5$ : $y_1=1+0.5(0+1)=1.5$ at $x=0.5$ .