Problems with dimensions when solving an ODE I'd like to solve the following differential equation:
$$\frac{dQ}{dt}=\frac{k_BT}{m}-\frac{\alpha Q}{m}$$ where $Q$ has units of $\text{m}^2\text{s}^{-1}$, $k_B$ is Boltzmann's constant, $T$ is temperature, $m$ is a mass and $\alpha$ has units $\text{kg s}^{-1}$.
Separating variables gives me $$\int\frac{dQ}{\frac{k_BT}{m}-\frac{\alpha Q}{m}}=\int dt$$
Hence $$-\frac{m}{\alpha}\ln\left(\frac{k_BT}{m}-\frac{\alpha Q}{m}\right)=t+C$$
Now, all of a sudden, I have something which does not hold up when I look at the dimensions: taking the logarithm of a dimensionfull quantity does not make any sense. Multiplying by $-\frac{\alpha}{m}$ and exponentiating yields $$\frac{k_BT}{m}-\frac{\alpha Q}{m}=e^{-\frac{\alpha t}{m}-C'}=C''e^{-\frac{\alpha t}{m}}$$ which is again problematic (left hand side has units $\text{m}^2\text{s}^{-2}$ while right hand side is unitless).
My question is: what am I missing here? This technique of solving ODE's is widely used, so I must be overlooking something simple. However, my professor was equally stumped as I am.
 A: When using this method of solving ODEs, we have to be careful not to use the indefinite integral.
$\int\frac{dQ}{\frac{k_BT}{m}-\frac{\alpha Q}{m}}=\int dt$ is actually $\int\limits_{Q_0}^Q\frac{dQ}{\frac{k_BT}{m}-\frac{\alpha Q}{m}}=\int\limits_{t_0}^t dt$
This gives us
$$-\frac{m}{\alpha}\ln\left(\frac{k_BT}{m}-\frac{\alpha Q}{m}\right)+ \frac{m}{\alpha}\ln\left(\frac{k_BT}{m}-\frac{\alpha Q_0}{m}\right)=t-t_0$$
As $\ln a - \ln b = \ln \frac{a}{b}$, this can be rewritten as 
$$-\frac{m}{\alpha}\ln\left(\frac{k_BT-\alpha Q}{k_BT-\alpha Q_0}\right)=t-t_0$$
Which has a dimensionless argument of the logarithm.
Another way of looking at it is that the constant $c$ you got after taking the indefinite integral was of the form $c_0-\ln(c_1)$, where $c_1$ has the same units as that of $\frac{\alpha Q}{m}$.
Usually, in mechanics (and most of physics), integrals are definite, even if we don't write them that way.
Also,while solving ODEs, it's a good idea to normalize everything to a dimensionless quantity before proceeding to solve. This isn't such a big deal for simple ODEs like the one above, but when dealing with complex coupled systems losing the units leads to fewer things to keep track of.
