Varying thermal conductivity with length

I'm interested in modeling a system where the material varies along its length, thus the conductivity coefficient would be a function of both T, and x. k(T,x). For starters, if I assume negligible change w.r.t T, then he heat diffusion equation would be d/dt(k(x)dT/dx)=0. Correct? What if k just equals x (ie. linear)

I'm a little rusty with this solution. Any help?

Thanks.