You are given the differential equation dy/dx = y' = f(x, y) with initial condition y(0 ) 1 = . The following numerical method is also given:
where f_{n} = f( x_{n} , y_{n}) = f(nh, y_{n}) x , y f nh, y = = and h is the integration step.
(a) Is this an explicit or implicit method? Explain you answer.
(b) Find an expression for local error of this method.
(c) If one takes f = - σy (where σ > 0 and constant) is this method stable? Prove your answer mathematically.
(d) What is the condition for the iterative convergence of the method (when f = - σy )? Show that fulfillment of the condition for stability of the method assures iterative convergence.