I have a linear partial differential equation with respect to $T(x,t)$:
$$\frac{\partial^2 T}{\partial x^2}-f(t)\frac{\partial T}{\partial x}=0,$$
which is subjected to two boundary conditions:
$T(h_1, t)=T_0(t)$, and $T(h_2, t)=T_{\infty}=const$, where $f(t)$ is an unknown function of $t$.
My question is can one regards it as a linear ordinary differential equation (DE) with constant coefficients. Thus, its characteristic equation can be
$$r^2-f(t)r=0.$$
So the two characteristic roots are $r_1=0, r_2=f(t)$, and the general solution to the homogeneous DE might be written as
$$T(x,t)=A \sinh [f(t)x]+B \cosh (0 x)=A \sinh [f(t)x]+B.$$
Thank you for your advice!