create a function that solves the bvp for an arbitrary number of points in matlab

To create a function that solves the boundary value problem (BVP) for an arbitrary number of points in MATLAB, you can use the built-in function bvp4c. Here's an example of how you can create the function:

main.m
function solution = solve_bvp(num_points)
% SOLVE_BVP solves the BVP for an arbitrary number of points

% Define the ODE system
odefun = @(x, y) [
    y(2); % y1' = y2
    -y(1)^2 + y(1) - x^2 + 2*x
];

% Define the boundary conditions
bcfun = @(ya, yb) [
    ya(1) - 0; % y1(a) - y1(0) = 0
    yb(1) - 1; % y1(b) - y1(1) = 1
];

% Define the mesh for the domain
x_mesh = linspace(0, 1, num_points);

% Solve the BVP
solinit = bvpinit(x_mesh, [0, 0]);
solution = bvp4c(odefun, bcfun, solinit);

% Plot the solution
figure;
plot(solution.x, solution.y(1, :))
xlabel('x')
ylabel('y_1')
title('Solution of BVP')

end
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In this example, the solve_bvp function takes a parameter num_points, which specifies the number of points to use in the discretization of the domain. The odefun defines the system of ODEs, and the bcfun defines the boundary conditions. The x_mesh defines the mesh for the domain, and we use bvpinit to initialize the solver. Finally, we use bvp4c to solve the BVP, and plot the solution.

You can call the solve_bvp function with the desired number of points to solve the BVP for that number of points, like this:

main.m
solve_bvp(100);
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This will solve the BVP using 100 equally spaced points in the domain [0, 1].