# Differential equation

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An equation containing one or more derivatives of a function is called a differential equation.

## Differential Equation

### 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

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

## Special nth derivative differential equation

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