Euler's Method

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John
John on 27 Mar 2011
Commented: Hiba Ahmed on 8 Dec 2017
Using the Euler method solve the following differential equation. At x = 0, y = 5.
y' + x/y = 0
Calculate the Numerical solution using step sizes of .5; .1; and .01
From my text book I have coded Euler's method
function [t,y] = eulode(dydt, tspan, y0, h)
%eulode: Euler ODE solver
% [t,y] = eulode(dydt, tspan, y0, h, p1, p2,...)
% ` uses EULER'S method to INTEGRATE an ODE
% (uses the slope at the beginning of the stepsize to graph the
% function.)
%Input:
% dydt = name of hte M-file that evaluates the ODE
% tspan = [ti,tf] where ti and tf = initial and final values of
% independent variables
% y0 = initial value of dependent variable
% h = step size
% p1,p2 = additional parameter used by dydt
%Output:
% t = vector of independent variable
% y = vector of solution for dependent variable
if nargin<4, error('at least 4 input arguments required'), end
ti = tspan(1); tf = tspan(2);
if ~ (tf>ti), error('upper limit must be greater than lower limit'), end
t = (ti:h:tf)';
n = length(t);
%if necessary, add an additional value of t
%so that range goes from t=ti to tf
if t(n)<tf
t(n+1) = tf;
n = n+1;
t(n)=tf;
end
y = y0*ones(n,1); %preallocate y to improve efficiency
for i = 1:n-1 %implement Euler's Method
y(i+1) = y(i) + dydt(t(i),y(i))*(t(i+1)-t(i));
end
plot(t,y)
I have made another m-file to run Eulode, what I am confused with is where do I input my different step sizes and where do I input x=0 and y=5. However since the analytical solution yields:
simplify(dsolve('Dy=-x/y','y(0)=5','x'))
ans =
(-x^2+25)^(1/2)
and when x=0 the value is 5 so I have coded my Euler's Method like the following and the final values are close to 5 so I think it is correct can someone just verify.
dydx=@(x,y) -(x/y);
[x1,y1]=eulode(dydx, [0 1],5,.5);
[x2,y2]=eulode(dydx,[0 1],5,.1);
[x3,y3]=eulode(dydx,[0 1],5,.01);
disp([x1,y1])
disp([x2,y2])
disp([x3,y3])
  1 Comment
Hiba Ahmed
Hiba Ahmed on 8 Dec 2017
what about if you have a system of 2 differential equations?

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Answers (1)

Walter Roberson
Walter Roberson on 27 Mar 2011
Yup, you have provided the values in the correct positions according to the documentation for eulode.
  2 Comments
John
John on 27 Mar 2011
I think it is correct too, but should the eulode not become more accurate with a smaller step size? (with this configuration .01 has a larger error than .1 and .5)
Matt Tearle
Matt Tearle on 28 Mar 2011
How are you determining the error? If you're using the calculation you used here http://www.mathworks.com/matlabcentral/answers/4165-plotting-error then that's an incorrect calculation. So your use of the code here is fine, and Euler's method is indeed more accurate with a smaller stepsize.

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