MATLAB: How to solve this

dsolve

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Following shows how to find a numerical solution. You ODEs seems stiff, therefore, I used a stiff solver (ode23s).

tspan = [0 10];
IC = [0; 0; 0];
[t, y] = ode23s(@odeFun, tspan, IC);
plot(t, y);
legend({'$\theta$', '$\dot{\theta}$', '$i(t)$'}, ...
    'FontSize', 16, 'Interpreter', 'latex', 'Location', 'best')
function dXdt = odeFun(t, X)
    J = 1;
    Kt = 0.1;
    b = 0.5;
    L = 1e-2;
    R = 10;
    V = 1;
    Ke = 0.5;
    
    % X is 3x1 vector
    % X(1) is theta, X(2) is theta_dot, X(3) = I
    
    theta = X(1);
    theta_dot = X(2);
    i = X(3);
    
    dXdt = zeros(3,1);
    dXdt(1) = X(2);
    dXdt(2) = 1/J*(Kt*i-b*theta_dot);
    dXdt(3) = 1/L*(-R*i+V-Ke*theta_dot);
end

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MATLAB: I want to add

Based on your comment on other question for expression of ea(t): https://www.mathworks.com/matlabcentral/answers/541922-please-tell-me-how-to-use-ode45-code#comment_883106 the following shows how to use ode45 to solve this ODE with input saturation

[t, y] = ode45(@odeFun, [0 1], [0; 0]);
plot(t, y(:,1), 'o-')
function dxdt = odeFun(t, x)
    % transfer function is equivalent to following ODE
    % x'' = 4/9(-16x'+ea(t))
    ea = (1-x(1))*303;
    if ea > 100
        ea = 100;
    elseif ea < -100
        ea = -100;
    end
    dxdt = zeros(2, 1);
    dxdt(1) = x(2);
    dxdt(2) = 4/9*(-16*x(2)+ea);
end

MATLAB: How to code simple pendulum motion using ode45

The equation of simple pendulum is

, which can be converted to two first order ODEs

and then using ode45 like this

theta_ic = [0.5; 0]; % initial conditions: theta(t=0)=0.5; dtheta(t=0)=0.
tspan = [0 10];
[t, theta] = ode45(@odeFun, tspan, theta_ic);
plot(t, theta);
legend({'$\theta$', '$\dot{\theta}$'}, ...
    'Location', 'best', ...
    'Interpreter', 'latex', ...
    'FontSize', 16)
function dtheta = odeFun(t, theta)
    g = 9.8;
    l = 1;
    % theta(1) = theta, theta(2) = dtheta
    
    dtheta = zeros(2, 1);
    dtheta(1) = theta(2);
    dtheta(2) = -g/l*theta(1);
end

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MATLAB: I want to add

MATLAB: How to code simple pendulum motion using ode45

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