function  f= Bernoulli(p,q,n)
% - Bernoulli first order nonlinear ode     y' +p(x) y= q(x) y^n
%  - is linearized by W =y^(1-n). to become W'+(1-n)*P(x)*W=(1-n)*q(x)
%  -  usage f=Bernoulli(p,q,n)
%
  syms x y c wh wp u
  disp(' Bernoulli type first order ode '); disp(q) 
%  disp(' for example you need:  syms x y p q w;p=x;q=1;n=-1; f=bernoulli(p,q,n ]'];
  disp(' You are solving  yprime+py=q y ^ n');
      disp(' where p = '); disp(p)
      disp(' and,  q = '); disp(q)      
      disp(' and, where n = '); disp(n)
  
     disp('Let  W = y^(1-n)')   
     disp('then the ode becomes linear nonhomogeneous ode as :') 
     disp(' Wprime +(1-n)pw =(1-n)q') 
   

      disp('Integrating Factor u= exp(int((1-n)*p))')
      
      u=simplify(exp(int((1-n)*p,'x')))
   
      disp(' Homogeneous solution is  wh=c/u')
      
      wh=c/u
      
      disp(' The Particular solution is wp=1/u * Int( (1-n)*u*q)')
      
      wp=simplify(int((1-n)*u*q)/u)
      
      disp(' The general  solution is w = wh+wp')
      
      
        w=simplify(wh+wp)
       y=w^(1/(1-n))
      disp(' y=w^(1/(1-n))=');disp(y)
      return