AHL 5.16 (HL)—Euler's method

For dy/dx=f(x,y), Euler's method uses y_(n+1)=y_n+h f(x_n,y_n). It replaces the curve on each short interval with the tangent at the current point.

Choose a step h, state the starting point and keep the same update rule. Smaller h usually reduces local error for a smooth problem, but accumulated error and model uncertainty remain.

For dy/dx=y, y(0)=1 and h=0.1, the first Euler step gives y₁=1+0.1(1)=1.1, while the exact value e^{0.1}≈1.105. The approximation is close but not identical.

Use the slope at the current point, not the next point, and do not confuse a numerical estimate with an exact solution. Check units of h and the direction of the step.

For a coupled system, update every component from the same current state: $x_{n+1}=x_n+h f_1(x_n,y_n,t_n)$ and $y_{n+1}=y_n+h f_2(x_n,y_n,t_n)$. Do not use the newly updated $x_{n+1}$ inside the same Euler step for $y$ unless a different method is explicitly specified. Technology or a spreadsheet may generate the table.