Are there any resources or examples of this method apart from what is described in the lecture notes?

I'm a bit confused. In the lecture notes we have

\begin{equation*}

u_{\alpha}(x,t)=\alpha u(\alpha x, \alpha^2 t).

\label{eq-4}

\end{equation*}

but in the question there is no $t$ term

\begin{equation*}

u_{\lambda}:= \lambda u(\lambda x, \lambda^\gamma )=u(x,\lambda)\qquad \forall \lambda>0

\end{equation*}