Question

help me please , By using MATLAB

How can I plot the solution of linear diffrential system ? I want the code in general case

Answer 1

Modified MATLAB Code:

close all
clear
clc

A = [1 0 1; 0 0 0; 0 0 -1]; % Coefficient Matrix
X0 = [5 7 6]'; % Initial condition

[V, D] = eig(A); % Get the Eigen values and Eigen vectors

% General Solution: X = Sum of {c_i * exp(L_i * t) * V_i}
% where c_i are constants, L_i are Eigen values and V_i are Eigen vectors
%
% At t = 0:
% X = X0 = Sum of {c_i * V_i}
% We need to solve the system V*C = X0, where C = [c_1, c_2, ..., c_i, ...]'
C = V\X0;

disp('Solution, X(t) =')
X = 0; syms t
for i = 1:length(A)
X = X + C(i) * exp(D(i,i)*t) * V(:,i); % Sum of {c_i * exp(L_i * t) * V_i}
end
X = simplify(X);
disp(X)

% Plotting
figure, hold on
t_vals = 0:0.001:2;
for i = 1:length(A)
plot(t_vals, subs(X(i), t_vals), 'DisplayName', sprintf('%s', X(i))) % Plot each solution
end
xlabel('t'), ylabel('X(t)'), title('Solution of ODE')
hold off
legend('Location', 'northwest')

Output:

Solution, X(t) =
8*exp(t) - 3*exp(-t)
7
6*exp(-t)

Plot:

Solution of ODE 8*exp(t) - 3*exp(-) 6*exp(-t) X(t) O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2