Arithmetic Course/Differential Equation/First Order Equation

First Order Equation
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
Equation is an expression of one variable such that
 * $$A \frac{d f(x)}{dx} + B f(x) = 0 $$
 * $$\frac{d f(x)}{dx} + \frac{B}{A} f(x) = 0 $$
 * $$\frac{d f(x)}{f(x)} = -\frac{B}{A} dx$$
 * $$\int \frac{d f(x)}{f(x)} = -\frac{B}{A} \int dx$$
 * $$Ln f(x) = -\frac{B}{A} t + C$$
 * $$f(x) = e^[-\frac{B}{A} t + C]$$
 * $$f(x) = e^C e^(-\frac{B}{A}) t $$
 * $$f(x) = A e^(-\frac{B}{A}) t $$