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##mx''+\beta x' + kx=f(x)## ; ##Lq''+Rq'+q/c=E(t)## (mass-spring; Kirchoff's diff eq.)

One proceeds with the math as if there were no actual units involved and just solved a problem dealing with only numbers.

Is there anyway to prove that there will always be dimensional validity without me having to go and see? I'm looking for some general argument.

This thing is causing me a huge headache and any help will be greatly appreciated.