 Solution

Use the stage solution continuous from $k_2=f(y)$ as $h\to 0$. Its displacement is $\delta =h[(1-a)f+a{k}_2]=hf+a{h}^2{f}^{\prime }f+O(h^3)$. Taylor expansion of $f(y+\delta )$ givesConsequently the Runge-Kutta method advances byThe exact solution has third-order term $h^3(f^{\prime 2}f+f^{\prime \prime }{f}^2)/6$. The first two orders agree for every real $a$, but the $f^{\prime \prime }{f}^2$ coefficient is always $1/4$ instead of $1/6$. Therefore the general nonlinear method has $\boxed{\text{order }2}$ for every fixed real $a$. The special value $a=1/3$ matches the linear test equation at third order but does not raise the nonlinear order. This is the two-stage Runge-Kutta family with an implicit second stage.