# Fundamental Mathematics/Arithmetic/Differential Equation

## Differential Equation

Equation has the general form

${\displaystyle A_{n}{\frac {d^{n}}{dt^{n}}}+A_{n-1}{\frac {d^{n-1}}{dt^{n-1}}}+\cdots +A_{1}{\frac {d}{dt}}+A_{0}=0}$

## Polynomial differential equation

### 1st ordered differential equation

Differential equation

${\displaystyle {\frac {d}{dx}}f(x)=sf(x)}$
${\displaystyle {\frac {df(x)}{f(x)}}f(x)=sdx}$
${\displaystyle \int {\frac {df(x)}{f(x)}}f(x)=\int sdx}$
${\displaystyle \ln f(x)=sx+c}$
${\displaystyle f(x)=e^{sx+c}=Ae^{st}}$ . With ${\displaystyle A=e^{c}}$

For 1st ordered differential equation of the general form

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$
${\displaystyle {\frac {d}{dx}}f(x)=-{\frac {B}{A}}f(x)}$
${\displaystyle sf(x)=-{\frac {B}{A}}f(x)}$
${\displaystyle s=-{\frac {B}{A}}}$
${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$

In summary,1st ordered differential equation of the general form

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

Solution of the 1st ordered differential equation above is

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

### 2nd ordered differential equation

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


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

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

## Special differential equation

Equation ${\displaystyle {\frac {d^{n}}{dt^{n}}}f(t)=-{\frac {1}{T}}f(t)}$ has roots ${\displaystyle f(t)=Ae^{st}=Ae^{\pm jn{\sqrt {\frac {1}{T}}}t}=ASin\omega t}$
with ${\displaystyle \omega =n{\sqrt {\frac {1}{T}}}}$

${\displaystyle {\frac {d^{n}}{dt^{n}}}f(t)=-{\frac {1}{T}}f(t)}$
${\displaystyle s^{n}f(t)=-{\frac {1}{T}}f(t)}$
${\displaystyle s=\pm jn{\sqrt {\frac {1}{T}}}f(t)}$
${\displaystyle f(t)=Ae^{st}=Ae^{\pm jn{\sqrt {\frac {1}{T}}}t}=ASin\omega t}$
${\displaystyle \omega =n{\sqrt {\frac {1}{T}}}}$