%%% Define the interval [a,b] and initial condition
a = 0;
b = 0.5;
y0 = 1;
%%% Find the approximated solution using Euler's method
N = 10; %%% number of steps
[x1,y1] = Euler(a,b,y0,N); %%% use the function "Euler" to find the approximate solution with step size h=0.05
N = 100; %%% number of steps
[x2,y2] = Euler(a,b,y0,N); %%% use the function "Euler" to find the approximate solution with step size h=0.005
%%% Define the exact solution
N = 100;
x3 = a:(b-a)/N:b;
y3 = 2*exp(x3) - x3 - 1;
%%% Plot the 3 solutions on the same figure
figure;
plot(x1,y1,'r'); hold on;
plot(x2,y2,'g'); hold on;
plot(x3,y3,'b'); hold off;
%%% Euler's method
function [x,y] = Euler(a,b,y0,N)
h = (b-a)/N;
x = a:h:b;
y = zeros(size(x));
f = @(x,y)x+y;
y(1) = y0;
%%% main loop for Euler's method
for i = 1:N
y(i+1) =y(i) + h * f(x(i),y(i));
end
end
