Bernoulli's Equation

Date: 12/7/95 at 10:18:33
From: Anonymous
Subject: Differential equation

Dear Dr. Math,

I'm trying to solve the following equation:

y'=a*y-b*y*y*y, y(0)=y_0

I believe this is an Riccati-differential-equation, but I don't know how 
to solve it.

Maybe you have a hint for me.

Greetings from Germany,

Andrea 

Date: 5/30/96 at 11:3:36
From: Doctor Anthony
Subject: Re: Differential equation

This is an example of Bernoulli's equation.

We have dy/dx = ay - by^3
    dy/dx - ay = -by^3    Divide through by y^3
  (1/y^3)dy/dx) - a/y^2 = -b

Make the substitution u = 1/y^2  So du/dx = (-2/y^3)(dy/dx) 
                                  (-1/2)du/dx = (1/y^3)dy/dx)

Substitute for y and dy/dx

     (-1/2)du/dx) - au = -b
           du/dx + 2au = 2b

Multiply by the integrating factor e^(INT(2a.dx))
                                = e^(2ax)

 e^(2ax).du/dx + 2au.e^(2ax) = 2b.e^(2ax)

    d/dx{u.e^(2ax)} = 2b.e^(2ax)

Integrate   u.e^(2ax) = 2b.INT{e^(2ax).dx}
            u.e^(2ax) = 2b(1/(2a)).e^(2ax) + const.

      (1/y^2).e^(2ax) = (b/a).e^(2ax) + const 

-Doctor Anthony,  The Math Forum