Hi @KG
If you show us the Euler method formula, then we don't have to search in Google or Wikipedia. This also ensures that the formula you give to us is correct and reliable with source cited. Anyhow, here is the demo. Hope that this is the Euler solution that you are looking for and acceptable.
function Demo_Euler
close all;
clc
tStart = 0;
step = 1e-2; % if step is smaller, then numerical solution is more accurate
tEnd = 1;
t = [tStart:step:tEnd]; % simulation time span
y0 = [0, 0]; % initial values
f = @(t,x) [x(2); % system of 1st-order ODEs
(t^3)*exp(2*t) - 4*x(1) - 4*x(2)];
yEuler = EulerSolver(f, t, y0); % calling Euler Solver
% analytical solution (can be obtained by hand, or by using 'dsolve')
yExact = (1/128)*exp(-2*t).*(3*(1 + t) + exp(4*t).*(-3 + 9*t - 12*t.^2 + 8*t.^3));
% compare numerical solution with analytical solution
plot(t, yEuler(1,:), t, yExact)
grid on
xlabel('Time, t')
ylabel('y_{1} and y_{2}')
title('Time responses of the system')
legend('Euler solution', 'Exact solution', 'location', 'best')
end
function y = EulerSolver(f, x, y0)
y(:, 1) = y0; % initial condition
h = x(2) - x(1); % step size
n = length(x); % number of steps
for i = 1:n-1
y(:, i+1) = y(:, i) + h*f(x(i), y(:, i));
end
end
Result:
