# 1.8 Multi-Step Methods

## 1.8.4 Backwards Differentiation Exercise

Measurable Outcome 1.12, Measurable Outcome 1.15, Measurable Outcome 1.18, Measurable Outcome 1.19

Suppose we wish to simulate the ODE \(u_ t = Au\) defined by the 2-by-2 matrix \(A\) for \({\Delta t}=0.1\). For which of the following \(A\) matrices will the fourth-order accurate backwards differentiation method be eigenvalue stable?

Exercise 1
&nbsp; \(A = \left[\begin{array}{cc} 0 & 1 \\ -180 & 0 \end{array}\right]\) &nbsp;
&nbsp; \(A = \left[\begin{array}{cc} 0 & 1 \\ -290 & -2 \end{array}\right]\) &nbsp;
&nbsp; \(A = \left[\begin{array}{cc} 0 & 1 \\ -8 & -4 \end{array}\right]\) &nbsp;
&nbsp; \(A = \left[\begin{array}{cc} -5 & 0 \\ 0 & 5 \end{array}\right]\) &nbsp;

Answer: The third matrix is the only one with both eigenvalues inside of the stability region for \({\Delta t}=0.1\).