1st ordered differential equation

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$ . Solution of the 1st ordered differential equation above is ${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$

${\displaystyle {\frac {d}{dx}}f(x)=-{\frac {B}{A}}f(x)}$

${\displaystyle sf(x)=-{\frac {B}{A}}f(x)}$

${\displaystyle s=-{\frac {B}{A}}}$

${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$

2nd ordered differential equation

${\displaystyle A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+C=0}$

${\displaystyle s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0}$

${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$

${\displaystyle \alpha =\gamma \beta ={\frac {R}{2L}}}$

${\displaystyle T=LC}$

${\displaystyle \gamma =RC}$

${\displaystyle A(\alpha )=Ae^{\alpha t}}$

${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$

One real root

${\displaystyle s=-\alpha }$ . ${\displaystyle I(t)=Ae^{-\alpha t}}$

Two real roots

${\displaystyle s=-\alpha \pm {\sqrt {\alpha -\beta }}}$ . ${\displaystyle I(t)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}}$

Two complex roots

${\displaystyle s=-\alpha \pm {\sqrt {\beta -\alpha }}}$ . ${\displaystyle I(t)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t}$