Solution
= Solution
Write the internal stage as
$$
k_2=f(y_n+h[(1-\theta)k_1+\theta k_2]).
$$
The two methods have $\theta=1/2$ and $3/4$. Expansion gives stability <function>
$$
R_\theta(z)=\frac{1+(1-\theta)z+(1/2-\theta)z^2}{1-\theta z}
=1+z+\frac{z^2}{2}+\frac\theta2z^3+O(z^4).
$$
Both therefore have order two; order three would require $\theta=1/3$.