Modeling and Simulation with ODE for MSE

(a) Show that for the solution x(t, x0) of the IVP with initial value x0 holds: ‖x(t, x0)‖ = ‖x(t0, x0)‖ . What is the geometric interpretation? Hint: Differentiate 1/2 ‖x(t)‖ and try to formulate the derivative as a scalar product! (b) Formulate an IVP in C (i.e. for complex-valued functions) equivalent (?) to the one in R in (a). (c) Consider now the numerical approximation x∆ to the IVP, where the approximation is calculated first by the explicit Euler method and second by the implicit Euler method, with constant step size h respectively. Is ‖x∆(t, x0)‖ constant for all t ≥ t0, too? Can you guess why it might be called the ’Martini glass effect’?