Question

Read the sample Matlab code euler.m. Use either this code, or write your own code, to solve first order ODE = f(t,y) dt (a). Consider the autonomous system Use Euler's method to solve the above equation. Try different initial values, plot the graphs, describe the behavior of the solutions, and explain why. You need to find the equilibrium solutions and classify them. (b). Numerically solve the non-autonomous system dy = cost Try different initial values, plot the graphs, describe the behavior of the solutions, and explain why. You should work on this in groups of 2 or 3 persons. Each group writes only one report. The report should follow the instruction

Answer 1

A)%%% Matlab code

clc;
close all;
clear all;
f=@(y) y^4-5*y^3+6*y^2;
t=0:0.01:2;
y(1,:)=[-2 -0.1 0 0.3 1.5];
for k=1:length(y(1,:))
for n=1:length(t)-1
y(n+1,k)=y(n,k)+0.01*f(y(n,k));
end
plot(t,y(:,k));
hold on
xlabel('t')
ylabel('y')

end
legend('y(0)=-2','y(0)=-0.1','y(0)=0','y(0)=0.3','y(0)=1.5');
%

OUTPUT:

2 y(0)-2 у(0)--0.1 1.5 y(0)=1.5 0.5 -0.5 -1 -2 0 0.2 0.40.6 0.8 1 1.2 1.416 1.8 2

(b)

clc;
close all;
clear all;
f=@(t,y)cos(t)*( y^4-5*y^3+6*y^2);
t=0:0.01:2;
y(1,:)=[-0.6 -0.1 0 0.3 0.8];
for k=1:length(y(1,:))
for n=1:length(t)-1
y(n+1,k)=y(n,k)+0.01*f(t(n),y(n,k));
end
plot(t,y(:,k));
hold on
xlabel('t')
ylabel('y')

end
legend('y(0)=-0.6','y(0)=-0.1','y(0)=0','y(0)=0.3','y(0)=0.8');
%

OUTPUT:

2 y(0)-0.6 y(0)-0.1 1.5 -y(0)-0.3 (0)=0.8 0.5 -0.5 -1 0 0.2 0.40.6 0.8 1 1.2 1.416 1.8 2