= Solution
Use sinusoidal <phasors>, $v(t)=\operatorname{Re}(\widetilde V e^{i\omega t})$. Differentiation becomes multiplication by $i\omega$, so the time-dependent element laws reduce to algebraic <complex impedances>:
$$
\boxed{Z_R=R,\qquad Z_L=i\omega L,\qquad Z_C=\frac1{i\omega C}.}
$$
Then $\widetilde V=Z\widetilde I$ encodes both amplitude ratio $|Z|$ and voltage-current phase difference $\arg Z$. Series impedances add, while parallel admittances add. This applies to linear sinusoidal steady state; switching transients are separate homogeneous solutions.
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