linearize a second-order system without symbolic math toolbox or linearize command in matlab

main.m
% Define parameters of the nonlinear system
m = 1;      % mass
k = 10;     % spring constant
b = 2;      % damping coefficient

% Define the differential equations
syms x(t) v(t)
eq1 = diff(x, t, 2) == -(k/m)*x - (b/m)*v;
eq2 = diff(v, t, 2) == -(k/m)*v - (b/m)*diff(x, t);

% Convert the differential equations to Laplace domain equations
X = laplace(x);
V = laplace(v);
EQ1 = subs(eq1, [x(t) diff(x(t), t) v(t) diff(v(t), t)], [X s*X V s*V]);
EQ2 = subs(eq2, [x(t) diff(x(t), t) v(t) diff(v(t), t)], [X s*X V s*V]);

% Approximate nonlinear terms using Taylor series expansion
EQ1 = taylor(EQ1, [X V], 'Order', 2);
EQ2 = taylor(EQ2, [X V], 'Order', 2);

% Substitute approximated equations back into Laplace domain equations
EQ1 = subs(EQ1, [X V diff(X, t) diff(V, t)], [x v s*x s*v]);
EQ2 = subs(EQ2, [X V diff(X, t) diff(V, t)], [x v s*x s*v]);
EQ1 = simplify(EQ1);
EQ2 = simplify(EQ2);

% Find the transfer function of the system in terms of the Laplace variable
H = solve(EQ1, V/s^2);

% Convert transfer function to state-space representation
[A,B,C,D] = tf2ss(double(coeffs(H, 'All')));
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