With an RLC series

${\displaystyle L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt+IR=0}$

${\displaystyle {\frac {d^{2}I}{dt}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0}$

${\displaystyle s=-(\alpha \pm {\sqrt {\alpha ^{2}-\beta ^{2}}})t=-(\alpha \pm \lambda )t}$

${\displaystyle I=e^{-}\alpha t(e^{\lambda }t+e^{-}\lambda t)}$

${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$

• ${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}=0.}$

${\displaystyle I=e^{-}\alpha t}$

• ${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}>0.}$

${\displaystyle I=e^{-}\alpha t(e^{\lambda }t+e^{-}\lambda t)}$

${\displaystyle I=ACos\lambda t}$

${\displaystyle A={\frac {e^{-}\alpha t}{2}}}$

• ${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}<0.}$

${\displaystyle I=A(e^{j}\lambda t+e^{-}j\lambda t)}$

${\displaystyle I=ASin\lambda t}$

${\displaystyle A={\frac {e^{-}\alpha t}{2j}}}$

The response of an LC series is a sinusoidal wave, or LC series can be used to produce decreasing sine wave oscillator.