Solution
= Solution
The chain rule gives
$$
\partial_x=\tau^{-p}\partial_\eta,\qquad
\partial_t=\partial_\tau-\frac{p\eta}{\tau}\partial_\eta.
$$
Choose
$$
p=\frac1{m+2},\qquad H(\tau)=\tau^{-1/(m+2)}.
$$
The PDE then reduces to the stated ODE. One integration, using symmetry at zero, gives $Y^mY'=-\eta Y$. Hence
$$
Y(\eta)=\left[\frac m2(1-\eta^2)\right]^{1/m},\qquad |\eta|\le1,
$$
which has the required endpoint values.