Here I have to show that the local truncation error for Ralston's method is $O(h^3)$ using a Taylor expansion of two variables, and then compare this with an appropriate taylor series method.
My attempt:
The taylor series expansion of two variables is:
$f(x+h,y+h)= f(x,y)+(h\frac{\partial}{\partial x}+k\frac{\partial}{\partial y})f(x,y)+\frac{1}{2}(h^2\frac{\partial^2}{\partial x^2}+2hk\frac{\partial^2}{\partial x\partial y}+k^2\frac{\partial^2}{\partial y^2})f(x,y)+O(h^3)\ (?)$
The local truncation error is then the error between the exact solution and the approximation in one step of the method... don't know what to do though
You need the Taylor expansion of $$y(x+h)=y(x)+f(x,y(x))h+\tfrac12(f_x+f_yf)h^2+...$$
Then compare with the Taylor expansions of \begin{align} k_1&=f(x,y),\\ k_2&=f(x+\tfrac23h,y+\tfrac23hk_1),\\ y_{+1}&=y+h(\tfrac14k_1+\tfrac34k_2). \end{align} Because of the factor $h$ in the last line you only need the linear expansion of the second line (the first does not contain any $h$) $$ k_2=f+f_x\,\tfrac23h+f_y\,\tfrac23hk_1+O(h^2)=f+\tfrac23(f_x+f_yf)h+O(h^2) $$ to confirm the identity up to order $O(h^3)$ with the Taylor expansion of $y(x+h)$, $$ y_{+1}=y+h(\tfrac14k_1+\tfrac34k_2)=y+fh+\tfrac34\tfrac23(f_x+f_yf)h^2+O(h^3). $$ If you want to compute the first coefficient of the error term, you need one degree more in the Taylor expansions.