# Fundamental Mathematics/Arithmetic/Differential Equation/2nd ordered differential equation

## 2nd ordered differential equation

2nd ordered differential equation has the general form

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

## Solving equation

${\displaystyle A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+Cf(x)=0}$
${\displaystyle {\frac {d^{2}}{dx^{2}}}f(x)+{\frac {B}{A}}{\frac {d}{dx}}f(x)+{\frac {C}{A}}f(x)=0}$
${\displaystyle s^{2}f(x)+{\frac {B}{A}}sf(x)+{\frac {C}{A}}f(x)=0}$
${\displaystyle s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0}$
${\displaystyle \beta ={\frac {C}{A}}}$
${\displaystyle \alpha ={\frac {B}{2A}}=\gamma \beta }$
${\displaystyle \gamma ={\frac {\alpha }{\beta }}={\frac {B}{2A}}{\frac {A}{C}}={\frac {B}{2C}}}$

The solution of the 2nd ordered polynomial equation above

 One real root ${\displaystyle s=-\alpha }$ ${\displaystyle i(t)=Ae^{-\alpha t}}$ Two real roots ${\displaystyle s=-\alpha \pm {\sqrt {\alpha -\beta }}}$ ${\displaystyle i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}}$ One complex roots ${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$ ${\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}}$

## Summary

Equation ${\displaystyle A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+Cf(x)=0}$ has roots
One real root . ${\displaystyle s=-\alpha }$ . ${\displaystyle i(t)=Ae^{-\alpha t}}$
Two real roots .  ${\displaystyle s=-\alpha \pm {\sqrt {\alpha -\beta }}}$ . ${\displaystyle i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}}$
One complex roots . ${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$ . ${\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}}$


## Application

RLC series circuit at equailibrium

${\displaystyle L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt+Ri(t)=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)+{\frac {R}{L}}{\frac {d}{dt}}i(t)+{\frac {1}{LC}}i(t)=0}$

Laplace transform yields Laplace equation below

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)+2\alpha {\frac {d}{dt}}i(t)+\beta i(t)=0}$
${\displaystyle s^{2}i(t)+2\alpha si(t)+\beta i(t)=0}$
${\displaystyle \beta ={\frac {1}{T}}}$
${\displaystyle T=LC}$
${\displaystyle \alpha ={\frac {R}{2L}}=\gamma \beta }$
${\displaystyle \gamma =RC}$

The roots of Laplace equation

 One real root ${\displaystyle s=-\alpha }$ ${\displaystyle i(t)=Ae^{-\alpha t}}$ Two real roots ${\displaystyle s=-\alpha \pm {\sqrt {\alpha -\beta }}}$ ${\displaystyle i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}}$ One complex roots ${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$ ${\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}}$