# Fundamental Mathematics/Arithmetic/Differential Equation/1st ordered differential equation

## 1st ordered differential equation

1st ordered differential equation has the general form of

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

## Find equation's root

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$
${\displaystyle {\frac {d}{dx}}f(x)=-{\frac {B}{A}}f(x)}$
${\displaystyle sf(x)=-{\frac {B}{A}}f(x)}$
${\displaystyle s=-{\frac {B}{A}}}$
${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$

## Summary

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$ . Solution of the 1st ordered differential equation above is ${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$