. solve the ode system below using • ode numerical solver • the symbolic toolbox • compare and visualize the solutions from both methods in a plot. dx dt = z + 4, x(0) = 2 dz dt = −3x, z(0) = 3 in matlab

To solve the given system of first-order ordinary differential equations using both the symbolic and numerical methods in MATLAB, you can follow these steps:

1. Using the Symbolic Toolbox:

   • Define the symbolic variables and the system of equations:

     main.m
     syms x(t) z(t)
     eq1 = diff(x) == z + 4;
     eq2 = diff(z) == -3*x;
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   • Define the initial conditions:

     main.m
     cond = [x(0) == 2, z(0) == 3];
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   • Solve the system symbolically:

     main.m
     sol_sym = dsolve([eq1, eq2], cond);
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2. Using the numerical solver:

   • Define a function that represents the system of equations:

     main.m
     function dxdt = ode_func(t, x)
         dxdt = [x(2) + 4; -3*x(1)];
     end
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   • Define the initial conditions:

     main.m
     x0 = [2; 3];
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   • Solve the system numerically using ode45:

     main.m
     tspan = [0 10];
     [t, sol_num] = ode45(@ode_func, tspan, x0);
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3. Compare and visualize the solutions:

   • Extract the symbolic solutions:

     main.m
     x_sym(t) = sol_sym.x;
     z_sym(t) = sol_sym.z;
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   • Plot the solutions:

     main.m
     figure;
     plot(t, sol_num(:, 1), 'b-', 'LineWidth', 2);  % Numerical solution for x
     hold on;
     fplot(x_sym, tspan, 'r--', 'LineWidth', 2);  % Symbolic solution for x
     plot(t, sol_num(:, 2), 'g-', 'LineWidth', 2);  % Numerical solution for z
     fplot(z_sym, tspan, 'm--', 'LineWidth', 2);  % Symbolic solution for z
     hold off;
     legend('Numerical solution for x', 'Symbolic solution for x', 'Numerical solution for z', 'Symbolic solution for z');
     xlabel('t');
     ylabel('Solution');
     title('Comparison of Numerical and Symbolic Solutions');
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This will solve the given system of ODEs using both the symbolic and numerical methods, and then plot and compare the solutions.