MATLAB

clear all; close all

a = 0; b = 0; c = 0; d = 0; 

J = [a b; c d]; 

eigs(J) 

t0 = 0; 

tf = 10; 

x0 = .1; 

y0 = .1; 

[t,u] = ode45(@(t,u) lab6fun(t,u,a,b,c,d), [t0 tf], [x0 y0]); 

x = u(:,1); 

y = u(:,2); 

%% 

set(0,'DefaultAxesFontSize',24) 

figure(1) 

hold on 

plot(x,y,'k','linewidth',3) 

plot(x0,y0,'go','MarkerSize',12,'MarkerFaceColor','g') 

xlabel('x') 

ylabel('y') 

figure(2) 

hold on 

plot(t,x,'b','linewidth',3) 

plot(t,y,'r--','linewidth',3) 

xlabel('t') 

legend('x','y') 

legend('boxoff') 

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function z = lab6fun(t,u,a,b,c,d)

x=u(1);

y=u(2);

z(1) = a*x + b*y;

z(2) = c*x + d*y;

z=z';

BASED ON THIS CODE , code up the forward euler method given the formula X_n+1=X_n+F(x_n)delta t

compare time courses of x(t) vs t from those using ODE45 and your coded up forward euler scheme from 2-4 , do these for different choices of delta t . Use the subplotcommand in MATLAB and create detailed legends so as to keep the total numbers of plots to a minimun.