Need help using ODE45 and eulers function

Alright so I have this ODE I solved for it using ODE45 and now its asking me to modify my script by using a eulers method script that was given to us and use eulers to solve for it now. I am lost. If its confusinf at all let me know and i can try to clarify it.

Here is my ODE file

function ex_with_2eqs
   t0 = 0; tf = 40; y0 = [-1;0];
   [t,Y] = ode45(@f,[t0,tf],y0,[]);
   y = Y(:,1); 
   v = Y(:,2); 
   figure(1);
   plot(t,y,'b+',t,v,'ro')  
   legend('y(t)','v(t)=y''(t)');
   title('Figure 1 L4g'); 
   grid on; 
   ylim([-1.5,1.5]); 
   figure(2); 
   plot(y,v); axis square; xlabel('y'); ylabel('v'); 
   title('Figure 2 L4g phase plot'); 
   grid on; 
   ylim([-1.5,1.5]);
   xlim([-1,1]);  
end

Here is my function file

function dydt = f(t,Y)
  y = Y(1); 
  v = Y(2);
  dydt = [ v; cos(t)-4*y^2*v-3*y ];
  end

and if it matters here is my eulers file although i know i dont have to do anything with it.

 f

unction [te,Ye] = euler(f,tspan,y0,N)

% Solves the IVP y' = f(t,y), y(t0) = y0 in the time interval tspan = [t0,tf]
% using Euler's method with N time steps.
% Input:
%  f     = name of inline function or function M-file that evaluates the ODE
%          (if not an inline function,  use: euler(@f,tspan,y0,N))
%          For a system, the f must be given as column vector.
%  tspan = [t0, tf] where t0 = initial time value and tf = final time value
%  y0    = initial value of the dependent variable. If solving a system,
%          initial conditions must be given as a vector.
%  N     = number of steps used.
% Output:
% t = vector of time values where the solution was computed
% y = vector of computed solution values.

m = length(y0);
t0 = tspan(1);
tf = tspan(2);
h = (tf-t0)/N;             % evaluate the time step size
t = linspace(t0,tf,N+1);    % create the vector of t values
y = zeros(m,N+1);   % allocate memory for the output y
y(:,1) = y0';              % set initial condition
for n=1:N
    y(:,n+1) = y(:,n) + h*f(t(n),y(:,n));   % implement Euler's method
end
t = t'; y = y';    % change t and y from row to column vectors
end

Answer 1

[Jonathan LeSage]

It looks to me that you're nearly there! I found a couple of small mistakes that seems to be holding you back.

Changes to your code:

• Replace ode45 with you defined euler function
• Read the documentation of your euler function. Unlike ode45 which is a variable step numerical solver, Euler's method is a fixed step solver. As such, you need to specify the number of steps you want to take, N, as the final fuction input. I would imagine 250+ steps would be sufficient for this problem.

Changes to your euler function:

• The two outputs of the function [te,Ye] are never defined in the code. From what I can tell, the outputs should be changes to [t,y].

Hope this helps to rectify the problem!