# Differential equations/Overview

## Differential Equation

For any differential equation has the general form

${\frac {d}{dt}}f(t)=f(t)$ Solving differential equation

${\frac {df(t)}{f(t)}}=dt$ $\int {\frac {df(t)}{f(t)}}=\int dt$ $Lnf(t)=t+c$ $f(t)=e^{t+c}=Ae^{t}$ with $A=e^{c}$ Similarly,

${\frac {d}{dt}}f(t)=sf(t)$ $f(t)=e^{t+c}=Ae^{st}$ with $A=e^{c}$ In general,

 Equation General Form Root of equation Differential equation ${\frac {d}{dt}}f(t)=f(t)$ $f(t)=e^{t+c}=Ae^{t}$ Differential equation ${\frac {d}{dt}}f(t)=-f(t)$ $f(t)=e^{-t+c}=Ae^{-t}$ Differential equation ${\frac {d}{dt}}f(t)=sf(t)$ $f(t)=e^{st+c}=Ae^{st}$ Differential equation ${\frac {d}{dt}}f(t)=-sf(t)$ $f(t)=e^{-st+c}=Ae^{-st}$ ### Ordered differential equations

 Equation General Form Root of equation 1st order differential equation ${\frac {d}{dt}}f(t)=-sf(t)$ $f(t)=e^{-st+c}=Ae^{-st}$ 2nd order differential equation ${\frac {d^{2}}{dt^{2}}}f(t)=-sf(t)$ $f(t)=e^{\pm j{\sqrt {s}}t+c}=Ae^{\pm j\omega t}=ASin\omega t$ . With, $\omega ={\sqrt {s}}$ nth order differential equation ${\frac {d^{n}}{dt^{n}}}f(t)=-sf(t)$ $f(t)=e^{\pm jn{\sqrt {s}}t+c}=Ae^{\pm j\omega t}=ASin\omega t$ . With, $\omega =n{\sqrt {s}}$ ## Ordinary differential equation

### 1st order ordinary differential equation

1st order ordinary differential equation of general form

$A{\frac {d}{dt}}f(x)+Bf(t)=0$ Rearrange equation above,

${\frac {d}{dt}}f(t)=-sf(t)$ . With, $s={\frac {B}{A}}$ Root of equation

$f(t)=Ae^{-st}=Ae^{-{\frac {B}{A}}t}$ ### 2nd order ordinary differential equation

$A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+Cf(x)=0$ ${\frac {d^{2}}{dx^{2}}}f(x)+{\frac {B}{A}}{\frac {d}{dx}}f(x)+{\frac {C}{A}}f(x)=0$ ${\frac {d^{2}}{dx^{2}}}f(x)=-{\frac {B}{A}}{\frac {d}{dx}}f(x)-{\frac {C}{A}}f(x)$ ${\frac {d^{2}}{dx^{2}}}f(x)=-2\alpha {\frac {d}{dx}}f(x)-\beta f(x)$ $\beta ={\frac {C}{A}}$ $\alpha =\beta \gamma ={\frac {C}{A}}\gamma ={\frac {B}{2A}}$ $\gamma ={\frac {B}{2C}}$ Roots of differential equations

• 1 real root . $f(x)=Ae^{-\alpha t}$ . $\alpha =\beta$ • 2 real roots . $f(x)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}$ . $\alpha >\beta$ • 2 complex roots . $f(x)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}$ . $\alpha <\beta$ $f(x)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t$ With

$A(\alpha )=Ae^{-\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$ 