If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$
Let $J$ be a non-empty open interval of $\mathbb { R }$. Does there exist a polynomial function that is a solution of (II.1) on $J$?