Improved Eulers Method Loop

I would like to use the improved eulers method to graph and solve the IVP y'=cot(y),y(0) = pi/6 using a step size of 1,0.5 and 0.25. My code currently does not generate a plot at I am not sure where there is a logic error. Could anyone point me in the right direction ?

% improved eulers method for stped size 1
y0 = pi/6;   
x0 = 0;
xn = 5;
h  = 1;  
dy = @(x,y) cot(y);  % dy/dx
y = y0;

for x = x0+h :h: xn
  y1 = y + dy(x,y)*h;
end 
plot(x,y1)
legend('Improved Euler 1')

hold on 

% Solve the ODE 

f = @(t,y) cot(y);
y0 = pi/6;
t2 = 0:1:5;

[t2,y2] = ode45(f,[0 5],y0);
plot(t2,y2,'LineWidth',3,'DisplayName','Exact solution');

legend show

Thanks,

MC

Answer 1

[Sulaymon Eshkabilov]

Hi,

You have overlooked the loop iteration and time space. Here is the corrected code:

% improved eulers method for stped size 1
y0 = pi/6;   
x0 = 0;
xn = 5;
h  = 1;      % h = 0.5; % h = 0.25;
dy = @(x,y) cot(y);  % dy/dx
y = y0;
x = x0:h:xn;
for ii = 1:numel(x)-1
y(ii+1) = y(ii) + dy(x(ii),y(ii))*h;
end 
plot(x,y, 'b-o')
legend('Improved Euler 1')
hold on 
% Solve the ODE 
f = @(t,y) cot(y);
y0 = pi/6;
t2 = 0:h:5;
[t2,y2] = ode45(f,t2,y0);
plot(t2,y2,'LineWidth',3,'DisplayName','Exact solution');
legend show
%%

Now you can extent the code with other time steps for h= 0.5 and 0.25 by changing the value of h.

Good luck.