= Solution
A constant-coefficient <linear multistep method>, normalized so that $\alpha_k=1$, has the form
$$
\sum_{j=0}^k\alpha_jY_{n+j}=h\sum_{j=0}^k\beta_jf(x_{n+j},Y_{n+j}),\qquad x_n=x_0+nh.
$$
Here $Y_n$ approximates the solution $y(x_n)$ of the <initial value problem>, and $Y_0,\ldots,Y_{k-1}$ 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.
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