Higher order: Taylor's series

We can go a step beyond Euler's method keeping up to second order terms in the expansion around $x_0$. Doing so we obtain

$$y(x+\Delta x)=y(x)+y'(x)\Delta x+\frac{1}{2}y''(x)(\Delta x)^2+O(\Delta x^3)$$ (5)

from (1) we get

$$y'(x)=f(x,y),$$ (6)

$$y''(x)=\frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial... ...l y}\frac{dy}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} f$$ (7)

Substituting in (5) we obtain

$$y_{n+1}=y_n+f\Delta x+\frac{1}{2}(\Delta x)^2[\frac{\partial f}{\partial x}+f\frac{\partial f}{\partial y}]+O(\Delta x^3),$$ (8)

where all the functions and derivatives are evaluated in $(x_n,y_n)$.