Module for the Bernoulli Differential Equation

Numerical Methods for O. D. E. using Mathematica. (c) John H. Mathews, 2003

Background

    A Bernoulli differential equation has the form  

        .  

Divide both sides by   and obtain  .  Then use the change of variable

      and  
    
to obtain

    .  

This latter D. E. is first order linear and can be solved using the method for first order linear D.E.'s.  
A function u[x] will result and the required the change of variables is again used to get a formula involving y[x].  

Transform the Bernoulli D.E. into a first order linear D.E.

Example 1.  Solve the Bernoulli differential equation  .