Euler's Method with Matrix

Sarah Johnson

2020 4 29

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I'm trying to implement Euler's Method on the following ODE, with initial condition [y1, y2] = [1, 3] and on the interval t in [0, 1]:

The exact solution is given as:

I currently have two different code as a way to go about solving this but I'm not sure which is correct/how to fix them. I'm really struggling to figure out where to go from here and any help/advice would be really appreciated!

% Code Number 1 %
function [y] = ForwardEulers(y0, t)
    n = 2;
    h = (t(2) - t(1)/n);
    y = y0;
    A = [-500.5, 499.5; 499.5, -500.5];
    [V, D] = eig(A);
    for t = 1:n
        y(t) = y0 * (V * ((eye(n) + h * (D))^t) * inv(V));
        x(t) = t * h;
    end
    plot(x, y)
end

% Code Number 2 %
function [y] = ForwardEulers2(y0, t)
    n = 2;
    h = (t(2) - t(1))/n;
    A = [-500.5, 499.5; 499.5, -500.5];
    for i = 1:n
        y(i) = ((eye(n) + (h* A))^i) * y0;
    end
end