Bernoulli's Equation

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