# Differential equation

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

## Differential Equation

### 1st ordered differential equation

$A{\frac {d}{dx}}f(x)+Bf(x)=0$ . Solution of the 1st ordered differential equation above is $f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}$ $A{\frac {d}{dx}}f(x)+Bf(x)=0$ ${\frac {d}{dx}}f(x)=-{\frac {B}{A}}f(x)$ $sf(x)=-{\frac {B}{A}}f(x)$ $s=-{\frac {B}{A}}$ $f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}$ ### 2nd ordered differential equation

$A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+C=0$ $s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0$ $\beta ={\frac {1}{T}}={\frac {1}{LC}}$ $\alpha =\gamma \beta ={\frac {R}{2L}}$ $T=LC$ $\gamma =RC$ $A(\alpha )=Ae^{\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$ One real root

$s=-\alpha$ . $I(t)=Ae^{-\alpha t}$ Two real roots

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

$s=-\alpha \pm {\sqrt {\beta -\alpha }}$ . $I(t)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t$ ## Special nth derivative differential equation

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