How to prove that $$ v(x,t) = x \cdot Du(x,t)+2tu_t(x,t)$ $$ is also a solution of the heat equation. $u_t(x,t)-\Delta u(x,t)=0$
Where $ u:R^{d \times1}\rightarrow R$ and $"\cdot" $ is dot product. $x\in R^d$ and $t\in R$
I have checked other answers where they used the fact that $u^\lambda = u(\lambda x, \lambda ^2 t)$ satisfy heat equation and that $v=\left.\frac{du^\lambda}{d\lambda}\right|_{\lambda=1}=v(x,t)$, I can see both facts are true. But I do not see how to use them to prove that $v$ satisfies the equation.
all kinds of solutions are highly appreciated.
Hint: $$(\partial_t - \Delta )u = 0 \implies \partial^\alpha (\partial_t - \Delta ) u = (\partial_t - \Delta )( \partial^\alpha u) = 0$$