Find the minimum value of $\int_0^1 y'^2 dx$ subject to the conditions $y(0)=y(1)=0$ and $\int_0^1y^2dx=1$.
In another question, I proved that, if we have an integral of the form
With end conditions $y(a)=y(b)=0$ and subject to the constraint $\int_a^b r(x)y^2dx=1$, the Euler-Lagrange equation becomes
Where $\lambda$ is our Lagrange multiplier. Now here, we have a similar integral, with $p=1$ and $q=0$. The above equation becomes
Which has the solution
Is this the minimum value for my integral? If it is, how can I solve for C and $\lambda$? I can use the constraint, but that would only give me one term.
Thank you in advance.