= Solution
Writing $f,f',f''$ at $y_n$, the implicit stage has the expansion
$$
k_2=f+h f'f+h^2\left(a(f')^2f+\frac12f''f^2\right)+O(h^3).
$$
Thus one step is
$$
y_{n+1}=y_n+hf+\frac{h^2}{2}f'f
+h^3\left(\frac a2(f')^2f+\frac14f''f^2\right)+O(h^4).
$$
The exact <Taylor theorem>[Taylor expansion] has third-order term
$$
\frac{h^3}{6}\bigl((f')^2f+f''f^2\bigr).
$$
The coefficients agree through order two for every real $a$, while the $f''f^2$ coefficient prevents order three for any $a$. Hence the <order of a Runge-Kutta method> is
$$
\boxed{2\quad\text{for every }a\in\mathbb R}.
$$
(The linear special case has one extra matched term when $a=1/3$, but the order for general nonlinear equations remains two.)
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