function y=linode20(a,b,c);
% - revision of march first 2002(RMM)
% - classifying a second order homogeneous ode with constant coefficients
% -    a y'' + b y'+ c y =0.
% -  usage y=linode20(a,b,c)
% -   to solve the nonhomogeneous equtation see linode22(a,b,c,q1)
   syms x m m1 m2 yh y1 y2 delta c1 c2 alpha beta
  disp(' Second order Linear homogeneous ODE with constant coefficients')
  disp(' a D2y + b Dy + C y= 0')

  
    sprintf(' where a = %d',a)
    sprintf('       b = %d',b)      
    sprintf(' and,  c = %d',c)
  
      disp('The characteristic equation is  a m^2+bm+c=0')
      %you could use solve here ...    solve('a*m^2+b*m+c', 'm');
         delta = b^2-4*a*c;
      sprintf(' delta= b^2-4ac =%d ',delta)
   
      if delta==0  
         disp(' class B (critical double roots m1=m2=-b/(2a)')
         m1=-b/(2*a);
         m2=m1;
         y1=exp(m1*x)
         y2=x*y1
      end
      
      if delta >0
            disp(' class A supercritical, two different real roots,')
            m1= (-b+sqrt(delta))/(2*a);
            m2= (-b-sqrt(delta))/(2*a);
            sprintf(' m1=%d',m1)
            sprintf('m2=%d',m2)
            y1=exp(m1*x)
            y2=exp(m2*x)
         end
         if delta <0
         disp(' class C subcritical , complex conjucate roots')
            alpha=-b/2/a
            beta=sqrt(-delta)/2/a
            y1=exp(alpha*x)*cos(beta*x)
            y2=exp(alpha*x)*sin(beta*x)
         end
         y=c1*y1+c2*y2;
         fprintf(' The Homogeneous solution yh=c1 y1+c2 y2 is :')
      return