The steady-state temperature distribution for a fin of cross-sectional area \(A\), constant perimeter \(P\), constant conductivity \(k\), and length \(L\) can be determined from 
the following differential equation:

\[
\frac{d}{d x}\left(k A \frac{d T}{d x}ight)-h P\left(T-T_{\infty}ight)=0
\]

where \(h\) is the heat transfer coefficient for the fin surrounded by a fluid with a constant temperature \(T_{\infty}\).

(a) Is this differential equation homogeneous or nonhomogeneous? With a change of variable \(\theta=T-T_{\infty}\), find the general solution for the variable \(\theta(x)\) by putting \(m^{2}=h P /(k A)\).

(b) Find a solution to the boundary value problem for the boundary conditions \(\theta(0)=\theta_{0}\) and \(\theta^{\prime}(L)=0\).