9.8. Consider the IVP for the ODE y = -y? with the initial condition y(0) = 1. We will use the backward Euler method to comp

9.8. Consider the IVP for the ODE y’ = -y? with the initial condition y(0) = 1. We will use the backward Euler method to compute the approxi- mate value of the solution yı at time ti = 0.1 (i.e., take one step using the backward Euler method with step size h = 0.1 starting from yo = 1 at to = 0). Since the backward Euler method is im- plicit, and the ODE is nonlinear, we will need to solve a nonlinear algebraic equation for yi. (a) Write out that nonlinear algebraic equation for yi. (6) Write out the Newton iteration for solving the nonlinear algebraic equation. (c) Obtain a starting guess for the Newton itera- tion by using one step of Euler’s method for the ODE. (d) Finally, compute an approximate value for the solution yı by using one iteration of Newton’s method for the nonlinear algebraic equation.



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