Arithmetic Course/Differential Equation/First Order Equation

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First Order Equation[edit]

The general form of First Order Equation

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

Which can be writte as

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

has one root of the exponential function form

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

Proof[edit]

Equation is an expression of one variable such that

A \frac{d f(x)}{dx} + B f(x) = 0
\frac{d f(x)}{dx} + \frac{B}{A} f(x) = 0
\frac{d f(x)}{f(x)} = -\frac{B}{A} dx
\int \frac{d f(x)}{f(x)} = -\frac{B}{A} \int dx
Ln f(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