## Description

1. Show that no choice of coeﬃcients in the one parameter family of the explicit two stage Runge-Kutta methods derived in class will result in the local error of order 4.

2. Consider the following method yn = yn−1 + h(θf(yn) + (1−θ)f(yn−1)), 0 ≤ θ ≤ 1, for solution of y0 = f(y).

(a) Compute the local error dn. For what value of θ, dn is the smallest? Why does this happen? (b) Find the values of θ for which the absolute stability region contains the whole left half plane of the complex plane?

3. Exercise 7.5 in Moler. It is not necessary to strictly follow the format of ode23tx. Note that Octave does not have ode113, so you do not have to consider it.