91Ó°ÊÓ

Convert the higher-order ordinary differential equation to a first-order system of equations. (a) \(y^{\prime \prime}-t y=0\) (Airy's equation) (b) \(y^{\prime \prime}-2 t y^{\prime}+2 y=0\) (Hermite's equation) (c) \(y^{\prime \prime}-t y^{\prime}-y=0\)

Show that the Implicit Trapezoid Method (6.89) is a second-order method.

Find a second-order, two-step implicit method that is weakly stable.

The Milne-Simpson Method is a weakly stable fourth-order, two-step implicit method. Are there any weakly stable third-order, two-step implicit methods?

(a) Use the matrix formulation to find the conditions on \(a_{i}, b_{i}\) required for a fourth-order, three-step implicit method. (b) Use the conditions to derive the Adams-Moulton Three-Step Method. (c) Show that the Adams-Moulton Three-Step Method is strongly stable.