Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

An explicit two-step method $y_{n+1} + a_1y_n + a_0y_{n-1} = h\left(b_1f_n + b_0f_{n-1}\right)$ has four free coefficients. Choose them to make the local order as high as possible (the method then reproduces every cubic solution exactly). The resulting method is useless in practice: find the largest modulus of a root of its characteristic polynomial $\rho(\xi) = \xi^2 + a_1\xi + a_0$.