Solving physics problems as a regular math problem. Why can we ignore units altogether?

Question

If I'm solving differential equations like

$Lq''+Rq'+\frac{q}{c}=V(t)$

or in general any kind of physics based problem.

I can proceed to solve it as if there were no units involved and all that was involved were the numbers.

I know experientially that I'll never lead to contradictions here, but the way I prove that this won't happen is by getting in to my method of solution and seeing how I didn't do anything silly. But this way of proving things is way too tedious, precisely because I'll have to consider each case.

Instead, I need a general argument that says that any problem that has units can be solved as an empty problem with no units and just numbers. Is there such an argument out there?

I'm sorry if my question is unclear but it's giving me a huge headache. Any help will be greatly appreciated.

Answer 1

You'll be fine up until you set boundary conditions. When those conditions have units, well, then you have to deal with units. Similarly, when you plug in constants, they will usually have units, so again, you'll have to deal with those. But if you stick with internally consistent units like MKS, you might get lucky.

Answer 2

Once you choose a consistent system of units, and express the variables and parameters in those units, the units should cancel and leave you an equation that involves only pure numbers. Otherwise, your equation is wrong. For example, an equation $A + B = 0$ where $A$ is a length and B is a mass can't be correct. Look up "dimensional analysis" for more on this.