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

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

Proof

Equation is an expression of one variable such that

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

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

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

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

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

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

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

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