UC Berkeley Math W128A: Programming Assignment 4 Prof. Per-Olof Persson (..n@berkeley.edu) Consider the initial-value problem y = f(t, y), a t b, y(a) = . (1) The Backward Euler method is given by w0 = (2) wi 1 = wi hf(ti 1, wi 1) (3) Since f depends on the unknown wi 1, (3) is an equation that must be solved at each time step, and in general f might be a non-linear function. In this assignment, we will implement the backward Euler method in MATLAB using Newton's method for solving (3). The solver will then be used to solve a sti dierential equation. 1. Write down Newton's method for solving (3) for wi 1, using the initial guess w (0) i 1 = wi . The iterations will depend on the partial derivative f /y = fy(t, y). Note: Your report needs to include this, even if your MATLAB codes are working correctly. Please write it using clear and concice mathematical notation. 2. Implement a MATLAB function backeuler.m of the form function [t,w] = backeuler(f, dfdy, a, b, alpha, N, maxiter, tol) which implements the backward Euler method using the Newton's method from problem 1