Past exam of the numerical linear algebra and numerical solution of differential equations course of the University of Oxford / 2020 / 2 / a / Solution

A constant-coefficient linear multistep method, normalized so that ${\alpha }_k=1$, has the formHere $Y_n$ approximates the solution $y(x_n)$ of the initial value problem, and $Y_0,\dots ,Y_{k-1}$ are starting approximations. The coefficients are independent of $n$ and $h$. If ${\beta }_k=0$, the update is explicit; otherwise it is an implicit time-stepping method, whose new value must satisfy the displayed equation.