Heat equation with variable dissapation I've been asked to solve the diffusion equation with variable dissipation, I have given the start of my answer but can't seem to proceed. Would appreciate a full solution as the work is due soon, thank you for any help though:
$$ \frac{∂u}{∂t} - D \frac{∂^2u}{∂x^2} + e^{-pt}u = 0 , x∈(-∞,∞), t∈[0,∞] $$
subject to 
$u(x,0) = φ(x),$
where $D>0$ and $p>0$ are given constants, and φ(x) is a given function.
I've managed to come up with this so far:  Let $u(x,t) = h(t)v(x,t)$ then
$$ h_tv + hv_t - Dhv_{xx} + e^{-pt}hv = 0 $$
$$  v_t - Dv_{xx} + \left(\frac{h_t}{h}+e^{-pt}\right)v = 0 $$
$v$ will satisfy the heat equation if the third term on the LHS is zero, i.e.
$$ \frac{h_t}{h} + e^{-pt} = 0 $$
$$ v_t - Dv_{xx} = 0 $$
with initial condition
$$ v(x,0) = \frac{u(x,0)}{h(0)} = \frac{\varphi(x)}{h(0)} $$

@DPH24 You can just choose your $h$ to be a solution of the first differential equation (the one for $h$ above). You can choose, for example,
$$
h(t)=e^{e^{-pt}/p}
$$
and then you solve for $v$ a standard IVP for the heat equation with $v(x,0)=e^{1/p}\phi(x)$. You get
$$
v(x,t)=\frac{e^{1/p}}{2\sqrt{\pi Dt}}\int\limits_{-\infty}^{+\infty}\phi(y)e^{-\frac{(x-y)^2}{4Dt}}\,dy
$$
And then $u=hv$.
