Arithmetic Course/Differential Equation/First Order Equation

First Order Equation

The general form of First Order Equation

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

Which can be writte as

${\frac {df(x)}{dx}}=-{\frac {B}{A}}f(x)$

has one root of the exponential function form

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

Proof

Equation is an expression of one variable such that

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

${\frac {df(x)}{dx}}+{\frac {B}{A}}f(x)=0$

${\frac {df(x)}{f(x)}}=-{\frac {B}{A}}dx$

$\int {\frac {df(x)}{f(x)}}=-{\frac {B}{A}}\int dx$

$Lnf(x)=-{\frac {B}{A}}t+C$

$f(x)=e^{[}-{\frac {B}{A}}t+C]$

$f(x)=e^{C}e^{(}-{\frac {B}{A}})t$

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