Answer to Question #220098 in Differential Equations for Anuj

Question

Write a program to solve the Cauchy-Euler differential equation

Answer 1

% Eulerian Method
% Setup and initial conditions
H = [0.25 0.1 0.05 0.01 0.005];  % step size
Y_euler = cell(1,numel(H));
for k=1:numel(H)
    h = H(k);
    x = 0:h:5;  % interval of x
    y = zeros(1, numel(x));  % allocate the resulting y
    y(1) = 1;  % inizial value of y
    n = numel(y);  % the number of y values
    % The loop that solves the differential equation:
    for i=1:n-1
        f = (y(i)-2*sin(x(i)));
        y(i+1) = y(i) + h * f;
    end
    Y_euler{k} = [x; y].';
end


syms Y(X)
eq = diff(Y) == Y - 2*sin(X);
ic = Y(0) == 1;
sol = dsolve(eq, ic);
y_sol = double(subs(sol, X, x));


hold on
plot(x, y_sol, 'r', 'DisplayName', 'true solution');
for k=1:numel(H)
    plot(Y_euler{k}(:,1), Y_euler{k}(:,2), 'b--', 'DisplayName', ['Euler method h=' num2str(H(k), '%.3f')]);
end
legend


err = zeros(1,numel(H));
for k=1:numel(H)
    err(k) = interp1(Y_euler{k}(:,1), Y_euler{k}(:,2), 2) - subs(sol, X, 2);
end