Laplace transform of Diffusion Equation

I've been using the Laplace transform to solve the above equation with certain boundary conditions. Ive done this successfully when D is constant. However, I have now figured out that D is infact not constant in the system im looking at, it depends on t. The exact form of D(t) is not known, but I know which value it starts for and that it decreases towards another value. My thinking was to choose a D(t) that made the problem solvable. My initial guess was:

which leads me to figure out

or

?

Is there any easy way to do this, for example some rules that let me isolate