Thank you!! Please let me know if this is better? Thanks very much just need hel

Question

Thank you!! Please let me know if this is better? Thanks very much just need help with exercise 1

Instructions: For your lab write-up follow the instructions of LAB 1. Euler's Method To derive Euler's method start from y(toyo and consider a Taylor expansion at tto h: y(ti) - y(to) +y'(to) (t1 -to)+... -yo hf (to, yo) +... For small enough h we get an approximation y for y(ti) by suppressing the ..., namely The iteration (L3.1) is repeated to obtain y2 ^ y(t2),... such that yn+1 yn + hf (tn, yn) = y(t1) Geometrically, the approximation made is equivalent to replacing theVi solution curve by the tangent line at (to, yo). From the figure we have y1 - yo from which (L3.1) follows As an example consider the IVP t1 C0 y'-4y-f (t, y) with y(0) 2 Note that here f does not explicitly depend on t (the ODE is called autonomous), but does implicitly through y -y(t). To apply Euler's method we start with the initial condition and select a step size h Since we are constructing arrays t and y without dimensionalizing them first it is best to clear these names in case they have been used already in the same MATLAB work session >> clear t y % no comma between t and y! type help clear for more info

Explanation / Answer

% DEFINING THE FUNCTION

function [t,y] = euler (f, tspan, y0, N)

m= length(y0);

t0= tspan(1);

tf= tspan(2);

h= (tf - t0)/N;

t= linespace(t0, tf, N+1);

y= zeros(m, N+1);

y(: ,1)= y0;

for k= 1: N

y(: , k+1)= y(: , k) + h*f( t(k) , y(: , k) );

end

t = t' ;

y = y' ;

end

% ACTUAL MATLAB CODE TO SOLVE THE EQUATION

f = inline('2*y' , 't' , 'y');

t= linespace(0, 0.5, 100);

y= 3* exp(2*t) ; % given equation

[t5, y5] = euler(f, [0,0.5], 3, 5); % calling function with N=5

[t50, y50] = euler(f, [0,0.5], 3, 50); % calling function with N=50

[t500, y500] = euler(f, [0,0.5], 3, 500); % calling function with N=500

[t5000, y5000] = euler(f, [0,0.5], 3, 5000); % calling function with N=5000

% PLOTTING THE RESULTS

plot(t5, y5, 'o-', t50, y50, 'x-', t500, y500, '*-', t5000, y5000, '@-', t, y, 'k-');

legend( 'Euler N= 5', 'Euler N= 50', 'EULER N= 500', 'EULER N= 5000',4);

% CALCULATING THE ERROR

e5= y(end) - y5(end);

e50= y(end)- y50(end);

e500= y(end)- y500(end);

e5000= y(end)- y5000(end);

%CALCULATING THE RATIOS OF THE ERRORS

ratio1= e5/e50;

ratio2= e50/e500;

ratio3= e500/e5000;

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