Question 1 asks if forward Euler is a single step method If you remember from the lecture, this is true In fact, forward Euler is the most basic of all single-step methods It takes exactly one step, and no interpolation is done Question 2 asks whether forward Euler and trapezoidal are members of the family of Runge-Kutta methods This is true Forward Euler is a first order Runge-Kutta method, and trapezoidal is a second order Runge-Kutta method So this is true Question 3 asks what symplectic ODE solvers are good for Symplectic ODE solvers are good for systems that are conservative That is, systems where you want to conserve energy, or area in the more abstract case With these systems, failure to use a symplectic ODE solver can sometimes inject friction, essentially You can imagine, with a pendulum for example, in the non-dissipative case, if your ODE solver is injecting a little bit of error every step size, every time you take a step, this could effectively act as a numerical friction term This loss of energy fails to conserve energy, and symplectic ODE solvers are developed specifically to keep the energy in the system Or the area, depending on the system youre working with