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taylor2.m

function [t,w] = taylor2(f,a,b,w0,N)

%

% Taylor's method of order 2 for y' = f(t,y) with y(a) = w0.

%

% Input:

% f function given in the above differential equation

% a left-end of [a,b]

% b right-end of [a,b]

% w0 initial value

% N number of steps

% Output:

% t vector of arguments

% w vector of approximate values of solution at t

h = (b - a)/N;

t = zeros(1,N+1);

w = zeros(1,N+1);

t(1) = a;

w(1) = w0;

for j=1:N,

tj = t(j);

yj = w(j);

D = feval('d2f',tj,yj);

w(j+1) = yj + h*(D(1)+h*(D(2)/2));

t(j+1) = a + h*j;

end

d2f.m

% Define and store the function d1 = f(t,y)

% and a formula for the first derivative

% of f(t,y) in the M-file d2f.m

function z = d2f(t,y)

d1 = y-t^2;

d2 = y-t^2-2*t;

z = [d1 d2];

taylor4.m

function [t,w] = taylor4(f,a,b,w0,N)

% Taylor's method of order 4 for y' = f(t,y) with y(a) = w0.

%

% Input:

% f function given in the above differential equation

% a left-end of [a,b]

% b right-end of [a,b]

% w0 initial value

% N number of steps

% Output:

% t vector of arguments

% w vector of approximate values of solution at t

h = (b - a)/N;

t = zeros(1,N+1);

w = zeros(1,N+1);

t(1) = a;

w(1) = w0;

for j=1:N,

tj = t(j);

yj = w(j);

D = feval('df',tj,yj);

w(j+1) = yj + h*(D(1)+h*(D(2)/2+h*(D(3)/6+h*D(4)/24)));

t(j+1) = a + h*j;

end

df.m

% Define and store the function d1 = f(t,y)

% and formulas for the next three derivatives

% of f(t,y) in the M-file df.m

function z = df(t,y)

d1 = y-t^2;

d2 = y-t^2-2*t;

d3 = y-t^2-2*t-2;

d4 = y-t^2-2*t-2;

z = [d1 d2 d3 d4];