Solving diffusion equation with variable dissapation I've been asked to solve the diffusion equation with variable dissipation:
$ \frac{∂u}{∂t} - D \frac{∂^2u}{∂x^2} + e^{-pt}u = 0 , x∈(-∞,∞), t∈[0,∞] $
subject to 
$u(x,0) = φ(x),$
where $D>0$ and $p>0$ are given constants, and φ(x) is a given function.
I think I need to use the substitution $u(x,t) = v(x,t)h(t)$ for some function h(t).  I would be grateful for a solution rather than hints. Thanks for any help. 

You're on the right track. Let's try to figure out what substitution will work. Let $u(x,t) = h(t)v(x,t)$ then
$$ h_tv + hv_t - Dhv_{xx} + e^{-pt}hv = 0 $$
$$ \implies v_t - Dv_{xx} + \left(\frac{h_t}{h}+e^{-pt}\right)v = 0 $$
$v$ will satisfy the heat equation if the third term on the LHS is zero, i.e.
$$ \frac{h_t}{h} + e^{-pt} = 0 $$
Can you take it from here?
Edit: Once you've found $h$ from the above ODE (a particular solution will do), then $v$ satisfies
$$ v_t - Dv_{xx} = 0 $$
with initial condition
$$ v(x,0) = \frac{u(x,0)}{h(0)} = \frac{\varphi(x)}{h(0)} $$
