The differential equation
is known as
Bernoulli's equation. If
n = 0, Bernoulli's equation reduces immediately to the standard form first-order linear equation:
If
n = 1, the equation can also be written as a linear equation:
However, if
n is not 0 or 1, then Bernoulli's equation is not linear. Nevertheless, it can be
transformed into a linear equation by first multiplying through by
y−
n
,
and then introducing the substitutions
The equation above then becomes
which is linear in
w (since
n ≠ 1).
Example 1: Solve the equation
Note that this fits the form of the Bernoulli equation with
n = 3. Therefore, the first step in solving it is to multiply through by
y−
n
=
y−3:
Now for the substitutions; the equations
transform (*) into
or, in standard form,
Notice that the substitutions were successful in transforming the Bernoulli equation into a linear equation (just as they were designed to be). To solve the resulting linear equation, first determine the integrating factor:
Multiplying (**) through the
yields
And an integration gives
The final step is simply to undo the substitution
w =
y−2. The solution to the original differential equation is therefore
Review and Introduction
First-Order Equations
