> y[2]:=x->(x-1)*(x-2)*(C[1]+C[2]*x);

             y[2] := x -> (x - 1) (x - 2) (C[1] + C[2] x)

> solve({int(x^2*diff(y[2](x),x)*diff(diff(y[2](x),x),C[1])+6*x*diff(y[2
> ](x),C[1]),x=1..2)=0,                                                 
>              
> int(x^2*diff(y[2](x),x)*diff(diff(y[2](x),x),C[2])+6*x*diff(y[2](x),C[
> 2]),x=1..2)=0},{C[1],C[2]}); 

                              -231         711
                      {C[2] = ----, C[1] = ---}
                              179          179

> 
