differential equation contains definite integral

Question

I am stuck on solving the following differential equation which contains a definite integral that I don't know how to deal with:

$$ f^{\prime\prime} + a^2 f - b\int_0^L f(t) \, dt = c$$

The boundary condition is $f(0)=0$ and $f(L)=R$. Anyone help me out of here? Thank you in advance.

Answer 1

Note that $\int_0^L f(t) dt$ is a constant depending on $L$ and call it $d$. Now you just need to integrate twice. I write $f(t)$ as $f$.

$$f' \frac{df'}{df}=c+bd-a^2f$$

Answer 2

Your solution is $$f(t)=A\cos(at)+B\sin(at)+C$$ where the boundary conditions and the original equation lead to 3 linear equations in the 3 parameters $A,B,C$, \begin{align} 0&=f(0)=A+C\\ R&=f(L)=A\cos(aL)+B\sin(aL)+C\\ c&=a^2C-b\left(\frac Aa\sin(AL)+\frac Ba(1-\cos(aL))+CL\right) \end{align} This linear system might be without solution, but then the problem is not solvable for the given boundaries and boundary conditions.