A simple question about the solution of homogeneous equation to this differential equation

Given that$t,1+t,{t}^{2},-t$ are the solutions to $y{}^{\u2034}+a\left(t\right)y{}^{\u2033}+b\left(t\right)y{}^{\u2033}+c\left(t\right)y=d\left(t\right)$ , what is the solution of homogeneous equation to this differential equation? What i have done is tried the properties of linear differential equation that

$L\left(t\right)=L(1+t)=L\left({t}^{2}\right)=L(-t)=d\left(t\right)$ so the homogeneous solution should be independent and i claim that $1,t,{t}^{2}$ should be the solution. However, i am not sure hot can i actually conclude that these are the solutions? It seems that it can be quite a number of sets of solution by the linearity.

Given that