Deriving Equations: Explain why $($using Kirchoff's rules$)$ equations $(3)$ and $(8)$ are true equations for LR and LC circuits, respectively.

Validating Solutions: Let's show that equations $(4)$ and $(9)$ are actually the behavior we expect for these circuits $($assuming the differential equations $(3)$ and $(8)$ hold$).$ Note that we're not actually deriving these formulas, just showing that they do work $($i$.$e$.,$ that they solve the differential equation like we want them to$)5$