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(a) Write Matlab/Mathematica routines implementing the basic step (x, y) → (x+h, ynext) in the case of Euler's method and Taylor's method of order p = 2, entering the function f of the differential equation y 0 = f(x, y) as an input function. (b) Consider the initial value problem y 0 = Ay, 0 ≤ x ≤ 1, y(0) = 1, where A = 1 2 λ2 + λ3 λ3 − λ1 λ2 − λ1 λ3 − λ2 λ1 + λ3 λ1 − λ2 λ2 − λ3 λ1 − λ3 λ1 + λ2 , 1 T = [111] Integrate the initial value problem with constant step length h = 1/N by (i) Euler's method (p = 1) and (ii) Taylor's method (p = 2) using the programs written in (a). Use N = 5, 10, 40, 80 and xn = .2 : .2 : 1, and suggested λ-values are: (i) λ1 = −1, λ2 = 0, λ3 = 1; (ii) λ1 = 0, λ2 = −1, λ3 = −10; (iii) λ1 = 0, λ2 = −1, λ3 = −40; (iv) λ1 = 0, λ2 = −1, λ3 = −160; (c) The exact solution is y(x) = −e λ1x + e λ2x + e λ3x e λ1x − e λ2x + e λ3x e λ1x + e λ2x − e λ3x Define the global error en = un − y(xn) based on (b). Print ||en||∞.

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function y_next=Taylorstep(x,y,h,f,f_x,f_y)
% y_next=Taylorstep(x,y,h,f,f_x,f_y)
% implements second order Taylor series method for ODE
% INPUTS:
% x -- x - scalar variable
% y -- y - vector variable (a column)
% h -- stepsize
% f -- f(x,y) -- differential equation
% f_x -- partial with respect to x of f
% f_y -- derivative of f with respect to y
% OUTPUT:
% y_next -- updated value of y

y_prime=f(x,y);

y_next=y+h*y_prime+(h^2/2)*(f_x(x,y)+f_y(x,y)*y_prime);

return...
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