Calculus Course/Differential Equations/1st Order Differential Equations

From Wikibooks, open books for an open world
Jump to: navigation, search

1st Order Differential Equations[edit]

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[edit]

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[edit]

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)