I have an equation on $f(x)$ which has the form $$\int_a^b K(x, y)f(y)dy = g(x) + f'(x=b)\times h(x).$$ The value of $f'(x=b)$ (first derivative of $f$, evaluated at $b$) is finite and given by a boundary condition on an antiderivative of $f$ (that derivative being a function of several variables); but to determine it, I first need to know $f$ to plug it in that boundary condition.
The kernel $K$ and the function $g$ and $h$ are series; the problem can be approximated by the linear system $$\mathbf{K}\mathbf f = \mathbf g + \alpha\mathbf h$$ where $\alpha = f'(x=b)$, for brevity.
My question is: can I legitimately treat $\alpha$ as a constant, solve the system so that $\mathbf f = {\mathbf{K}}^{-1}\bigl(\mathbf g + \alpha\mathbf h\bigr)$, and plug the solution into my boundary condition to determine the value of $\alpha$, leading to a final expression for $f$? Or does the fact that $\alpha$ is related to my unknown function forbids me to do that?