function  y2= ROO(p,q,y1)
% - March 16,2002 revision(RMM)
% - ROO is the reduction of order  for D2y+p*Dy+q*y=0.
% - P and Q are functions of x.
% - y1 is one solution and ROO will find the second linearly indep. solution y2. 
% -
% -                    /
% -                    | - p  dx
% -          /   exp(  /         )
% -  y2=  y1 |  ------------------  dx
% -          /       (y1)^2
% -
% -
% -      usage y2=ROO(p,q,y1)

syms x 
disp(' Reuction of order ')
disp(' You know y1(x) is a solution for D2Y+P*Dy+q*y=0  ')
disp(' Find y2=y1(x)*u(x). ')
            disp(' where p(x) = '); disp(p)
            disp(' and,  q(x) = '); disp(q) 
%  -  let us also verify that y1 is a solution.            
z1=diff(diff(y1,'x'),'x')+p*diff(y1,'x')+q*y1;
z1=simplify(z1)
    if z1==0 
        disp ('Tested y1 is a solution. So we proceed to find y2')
         z2=int(p,x)
         z2=exp(-z2)
         z2=simplify(z2/y1/y1)
         u=simplify(int(z2,x))   
         disp(' The second linearly inependent solution y2 is= ')    
         y2=simplify(u*y1)
         disp( 'Extra the wronskian w= y1*Dy2-y2*Dy1='); 
         w=y1*diff(y2,'x')-y2*diff(y1,'x');
      else
         disp ('z1=');disp(z1)
         disp(' So y1 is not a solution. try again')
     end
     return