I have a problem deriving the following formula:
$$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$
Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ where $A_0$ is the value of $A$ when $t=0$.
The problem lies in the fact that it $P/T$, so I'm having a little trouble with that.
If I take the figures of your comment, then the (linear) function for the temperature is
$T(h)=-\frac{20}{3}h+290$
This expression can be inserted in your differential equation.
$$\frac{dP}{dh} = k\left(\frac{P}{-\frac{20}{3}h+290}\right)$$
Now you can seperate the variables and then integrate both sides.