I would like to solve the following Diriclet eigenvalue problem $(*)$ on interval $[0,a]$ \begin{equation} \begin{cases} f^{''}+\lambda f=\frac{1}{a}\int_0^a f^{''}\ dt\ \ :=H\\ f(0)=f(a)=0 \end{cases} \end{equation} If we look at the standard problem: \begin{equation} \begin{cases} f^{''}+\lambda f=0\\ f(0)=f(a)=0 \end{cases} \end{equation} The solutions are $$\left(\frac{k^2\pi^2}{a^2},\sin(\frac{k\pi}{a}x)\right), k=1,2,\cdots$$
I found that when $k$ is even, the solution is also solution of the problem $(*)$, i.e., $$\left(\frac{4k^2\pi^2}{a^2},\sin(\frac{2k\pi}{a}x)\right), k=1,2,\cdots$$ since the intergration of even parts are all zeros. So I wonder are these only solutions of $(*)$? I tried to write down the general solution of problem $(*)$ using formula from ODE, I got something like $$f=c_1\sin(\sqrt{\lambda}x)+c_2\cos(\sqrt{\lambda}x)+\int_0^x\frac{\sin(\sqrt{\lambda}s)\cos(\sqrt{\lambda}x)-\sin(\sqrt{\lambda}x)\cos(\sqrt{\lambda}s)}{-\sqrt{\lambda}}\ H ds$$ Honestly I am not sure if I can use the formula for nonhomogeneous ODE since the RHS depends on $f$. Even I can use it, when I plug in boundary conditions, things get messy.
Could anyone give some hints? Very appreciated.