I'm looking for a function $y(x)$ which minimizes following equation $$ \int_0^L(y''(x)-f(x))^2dx $$ under the constraints that $$ \int_0^Ly(x)dx=A\\ y(x\in[0,L])>0 \\ y(0)=0\\ y(L)=0\\ $$ with $f(x)$ being known function that I will be plugging in later, however, I can't figure what equations I should be solving. If I understand correctly the Lagrange multipliers can't be used here, as my condition is for the value of the integral, and not for the value of the function for every given $x$.
Could someone point me in the right direction about what I should be reading about?
You need to find the extremals of the functional $$ \int\limits_0^L(y''(x)-f(x))^2+\lambda y(x))\,dx $$ For a reference, you can check Gelfand-Fomin, section 12. (page 43 in my edition)
Observe that here $\lambda$ is a constant, not a function of $x$ as it is the case when you have finite (non-integral) subsidiary conditions.