I’m a mathematician and i’m working with the brachistochrone with friction equation, i.e., the optimization problem $$operatorname{min}int_0^1 sqrt{frac{1+y'(x)^2}{y(x)-mu x}}dx,$$ with $y(0)=0$ and $y(1)=1$. I have obtained the analytic solution but im asked to: Discretize the variational problem by taking different subintervals and solve the approximate problem using some commercial software (for example, using the fmincon function in Matlab). Compare with the exact solution. What do you observe as the discretization becomes finer?. The exact solution is begin{align*}
x(theta)&=frac{C}{2}left(theta-sintheta+mu(1-costheta)right),
y(theta)&=frac{C}{2}left(1-costheta+mu(theta+sintheta)right),
end{align*} I’ve never used fimcon before and im not able to understand how it works. Could you help me? Thanks in advance!

The exact solution is $$
x(theta)=frac{C}{2}left(theta-sintheta+mu(1-costheta)right),
y(theta)=frac{C}{2}left(1-costheta+mu(theta+sintheta)right). $$ I’ve never used fimcon before and im not able to understand how it works. I tried to reproduce some examples found on matlab but they are not this type of optimization problem. Could you help me? Thanks in advance!