Substitute the exact solution and expand every value about $t_n$. The coefficient of $h^q y^{(q)}(t_n)$ in the defect is zero for $q=0,\ldots,p$ exactly when
$$
\boxed{\sum_{k=0}^s\rho_k=0,\qquad
\sum_{k=0}^sk^q\rho_k
=q\left[s^{q-1}\sigma_s+(s-1)^{q-1}\sigma_{s-1}\right]
\quad(1\leq q\leq p).}
$$
These are the <order conditions for a linear multistep method>, specialized to the two nonzero derivative coefficients. The first failed identity determines the leading <local truncation error>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section A / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-341/section-a/1/a/solution