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