Arithmetic Course/Differential Equation/First Order Equation

First Order EquationEdit

The general form of First Order Equation

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

Which can be writte as

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

has one root of the exponential function form

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

ProofEdit

Equation is an expression of one variable such that

${\displaystyle A{\frac {df(x)}{dx}}+Bf(x)=0}$
${\displaystyle {\frac {df(x)}{dx}}+{\frac {B}{A}}f(x)=0}$
${\displaystyle {\frac {df(x)}{f(x)}}=-{\frac {B}{A}}dx}$
${\displaystyle \int {\frac {df(x)}{f(x)}}=-{\frac {B}{A}}\int dx}$
${\displaystyle Lnf(x)=-{\frac {B}{A}}t+C}$
${\displaystyle f(x)=e^{[}-{\frac {B}{A}}t+C]}$
${\displaystyle f(x)=e^{C}e^{(}-{\frac {B}{A}})t}$
${\displaystyle f(x)=Ae^{(}-{\frac {B}{A}})t}$