Fundamental Mathematics/Arithmetic/Differential Equation

Differential Equation

Equation has the general form

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

Polynomial differential equation

1st ordered differential equation

Differential equation

{\frac {d}{dx}}f(x)=sf(x)

{\frac {df(x)}{f(x)}}f(x)=sdx

\int {\frac {df(x)}{f(x)}}f(x)=\int sdx

\ln f(x)=sx+c

f(x)=e^{sx+c}=Ae^{st}

. With

A=e^{c}

For 1st ordered differential equation of the general form

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}

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

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

Solution of the 1st ordered differential equation above is

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

2nd ordered differential equation

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

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

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

s^{2}f(x)+{\frac {B}{A}}sf(x)+{\frac {C}{A}}f(x)=0

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

\beta ={\frac {C}{A}}

\alpha ={\frac {B}{2A}}=\gamma \beta

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

Special differential equation

Equation  ${\frac {d^{n}}{dt^{n}}}f(t)=-{\frac {1}{T}}f(t)$  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(t)

f(t)=Ae^{st}=Ae^{\pm jn{\sqrt {\frac {1}{T}}}t}=ASin\omega t

\omega =n{\sqrt {\frac {1}{T}}}

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