I have a coding project where I have to solve an equation that could be simplified as:
$$ \sum_{i=1}^{m} \sin(i*x)*f(C_i)=0, \forall x \in [0,\pi], $$
where I have to find all the $C_i$ (not $x$), $C_i$ does not depend of $x$ only of $i$, and because of that I assumed that I hat to solved $f(C_i)=0$ equations, but my teacher said that I was wrong and to get the $m$ equations needed I have to substitute the first equation with $m x$ (excluding the trivial $\pi$ and $0$). And then solve.
After programing, for my surprise, I got the same solutions.
Is my solution valid?
Since maybe i´m not the best explainer, i put you in context, I have to program a levy plate deformation solver. the problem is rule by this equation:
$ u_z(x,y)= \sum_{i=1}^{m}\sin({\pi ix\over a})*(A_i\cosh({\pi iy\over a})+B_i\sinh({\pi iy\over a})+{\pi iy\over a}(C_i\cosh({\pi iy\over a})+D_i\sinh({\pi iy\over a}))+f_i(y)), a\in \mathbb{R} $
It must follow the following the boundary conditions
$ \forall x,\in [0,a],\ and\ y=\pm{b\over 2},\ \frac{\partial^2 u_z}{\partial y^2}+\nu\frac{\partial^2 u_z}{\partial x^2}=0, \ and \ \frac{\partial^3 u_z}{\partial y^3}+(2-\nu)\frac{\partial^3 u_z}{\partial x^2\partial y}=0 $
After playing a bit with the equations you end up with 4 equations similar to the simplify version above. Since x is linearly independence of the coefficients $A_i\ B_i\ C_i\ D_i $, i think just by solving the 4 $ f(C_i)$ for each i is sufficient enough, but maybe I'm wrong. I tried expanding the $sin(nx)$ series but I lack mathematical knowledge to follow that route. Because of that I lack the knowledge to proof me or my teacher wrong. When I asked my teacher about this his explanation didn't seem to give a hint of why I was wrong, so I write in their forum to get a certain answer because withe the two versions of the solver I have, it doesn't matter what I do to prove me wrong (increasing m, $f_i(y)$
Thanks beforehand