First note that the diffusion p.d.e. is where k is a constant (the thermal diffusivity).
Now note that if then (using the product rule):
Substitute all that lot into and divide through by the common (non-zero) factor of :
Comparing this with the diffusion equation, you clearly require the coefficients of and v to equal zero:
Substitute (1) into (2) and solve for w:
There are probably numerous typos in the above - work through it carefully, double check everything .....