Calculus Course/Differential Equations/1st Order Differential Equations

1st Order Differential Equations
A 1st order differential equation has the form shown below
 * $$A \frac{d}{dx} f(x) + B = 0$$

It can be shown that roots o the differential equation above is
 * $$f(x) = Ae^(- \alpha x) $$
 * $$\alpha = \frac{B}{A}$$

Proof
The above equation can be rewritten as
 * $$\frac{d f(x)}{dx} + \frac{B}{A} f(x) = 0$$

Then
 * $$\frac{d f(x)}{dx} = - \frac{B}{A} f(x) $$
 * $$\int \frac{d f(x)}{f(x)} = - \frac{B}{A} \int dx$$
 * $$Ln f(x) = - \frac{B}{A} x + C$$
 * $$f(x) = e^(- \frac{B}{A} x + C)$$
 * $$f(x) = Ae^(- \frac{B}{A} x) $$

Summary
First ordered differential equation of the form
 * $$A \frac{d}{dx} f(x) + B = 0$$

has a exponential root of the form
 * $$f(x) = e^(-\alpha x + c)$$

where
 * $$\alpha x = \frac{B}{A}$$
 * $$A = e^c$$

or
 * $$f(x) = A e^(-\alpha x)$$