This paper by G. G. Dahlquist discusses a special stability problem for linear multistep methods. The trapezoidal formula is noted for having the smallest truncation error among linear multistep methods with a certain stability property. Error bounds for this method are derived under general conditions. To ensure that the error remains bounded as $ t \rightarrow \infty $, even with a large product of the Lipschitz constant and the step-size, it is only necessary to assume that the integral curve is uniformly asymptotically stable in the Liapunov sense. The paper introduces the general linear k-step method for solving first-order ordinary differential equations. The method is defined by a formula involving constants $ \alpha_i $ and $ \beta_i $, and a step-size $ h $. The theory of such methods is well-established, as discussed in Henrici's book. The method is explicit when $ \beta_k = 0 $, and implicit otherwise, requiring additional conditions on $ h $ and $ f $ for existence and uniqueness. The paper defines polynomials $ \varrho(\zeta) $ and $ \sigma(\zeta) $, and an operator $ L $, which is used to determine the order of the method. The order is the largest integer $ p $ such that $ L\varphi(t) = 0 $ for all polynomials $ \varphi(t) $ of degree $ p $. The method is consistent if $ p \geq 1 $, and the error constant $ c^* $ is used to compare the accuracy of methods with the same order. It is known that for a given $ k $, the maximum order $ p $ is $ 2k $. However, stability requirements reduce this maximum order. For instance, no method with $ p > k + 2 $ can possess a certain stability property, which is reasonable for extensive numerical integration. The paper highlights the importance of stability in ensuring that the numerical solution remains close to the true solution over long time intervals.This paper by G. G. Dahlquist discusses a special stability problem for linear multistep methods. The trapezoidal formula is noted for having the smallest truncation error among linear multistep methods with a certain stability property. Error bounds for this method are derived under general conditions. To ensure that the error remains bounded as $ t \rightarrow \infty $, even with a large product of the Lipschitz constant and the step-size, it is only necessary to assume that the integral curve is uniformly asymptotically stable in the Liapunov sense. The paper introduces the general linear k-step method for solving first-order ordinary differential equations. The method is defined by a formula involving constants $ \alpha_i $ and $ \beta_i $, and a step-size $ h $. The theory of such methods is well-established, as discussed in Henrici's book. The method is explicit when $ \beta_k = 0 $, and implicit otherwise, requiring additional conditions on $ h $ and $ f $ for existence and uniqueness. The paper defines polynomials $ \varrho(\zeta) $ and $ \sigma(\zeta) $, and an operator $ L $, which is used to determine the order of the method. The order is the largest integer $ p $ such that $ L\varphi(t) = 0 $ for all polynomials $ \varphi(t) $ of degree $ p $. The method is consistent if $ p \geq 1 $, and the error constant $ c^* $ is used to compare the accuracy of methods with the same order. It is known that for a given $ k $, the maximum order $ p $ is $ 2k $. However, stability requirements reduce this maximum order. For instance, no method with $ p > k + 2 $ can possess a certain stability property, which is reasonable for extensive numerical integration. The paper highlights the importance of stability in ensuring that the numerical solution remains close to the true solution over long time intervals.