Fundamental Mathematics/Arithmetic/Differential Equation/2nd ordered differential equation

2nd ordered differential equation

2nd ordered differential equation has the general form

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

$A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+Cf(x)=0$ ${\frac {d^{2}}{dx^{2}}}f(x)+{\frac {B}{A}}{\frac {d}{dx}}f(x)+{\frac {C}{A}}f(x)=0$ $s^{2}f(x)+{\frac {B}{A}}sf(x)+{\frac {C}{A}}f(x)=0$ $s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0$ $\beta ={\frac {C}{A}}$ $\alpha ={\frac {B}{2A}}=\gamma \beta$ $\gamma ={\frac {\alpha }{\beta }}={\frac {B}{2A}}{\frac {A}{C}}={\frac {B}{2C}}$ 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 $s=-\alpha \pm j{\sqrt {\beta -\alpha }}$ $i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}$ Summary

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

RLC series circuit at equailibrium

$L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt+Ri(t)=0$ ${\frac {d^{2}}{dt^{2}}}i(t)+{\frac {R}{L}}{\frac {d}{dt}}i(t)+{\frac {1}{LC}}i(t)=0$ Laplace transform yields Laplace equation below

${\frac {d^{2}}{dt^{2}}}i(t)+2\alpha {\frac {d}{dt}}i(t)+\beta i(t)=0$ $s^{2}i(t)+2\alpha si(t)+\beta i(t)=0$ $\beta ={\frac {1}{T}}$ $T=LC$ $\alpha ={\frac {R}{2L}}=\gamma \beta$ $\gamma =RC$ The roots of Laplace equation

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