The material of this question is relevant to solving boundary value problems, Rayleigh-Benard convection and the Boussinesq equations.
How might I solve the boundary value problem ( BVP ),defined by
\begin{align} \left[-S+ \left( \frac{d^2}{dx^2} -k^2\right)\right]\circ \left( \frac{d^2}{dx^2} -k^2\right)f(x)&=0 \tag1\\ f(0)&=0\\ f(h)&=\Delta\theta \end{align} with $k$ constant.
NB: In context I like the use of the notation \begin{align} D&= \frac{d}{d x}\\ D^2&= \frac{d^2}{d x^2}\\ \end{align} Using this notation (1) can be written $$[-S+(D^2-k^2)]\circ(D^2-k^2)f(x)=0 \tag2$$
The ordinary differential equation (2) is of fourth order and is homogenous.
OTHER INFORMATION
I searched on this site and found a mention of the phrase ' undetermined coefficients ' , but I don't think that was in relation to the BVP idea, I can't remember.
I was hoping for some specific help with details.
Without more information, $S$ is also a constant and the roots of the characteristic equation are
$$\pm k,\pm\sqrt{S+k^2}.$$
Hence the general solution
$$f(x)=C_0e^{kx}+C_1e^{-kx}+C_2e^{\sqrt{S+k^2}x}+C_3e^{-\sqrt{S+k^2}x}.$$
The given initial conditions yield
$$C_0+C_1+C_2+C_3=0.$$
$$C_0e^{kh}+C_1e^{-kh}+C_2e^{\sqrt{S+k^2}h}+C_3e^{-\sqrt{S+k^2}h}=\Delta\theta.$$
Two more conditions are necessary to determine all coefficients.