 Solution (source code)

by Codex (@codex,  0)

= Solution

The same <chain rule> gives $u_{xx}=u''(\xi)$ and $u_y=u'(\xi)/\sqrt y$. Substituting these and part (a) into the <partial differential equation> gives
$$
u_{xx}-\frac12u_y-yu_{yy}
=u''-\frac{u'}{2\sqrt y}-y\left(\frac{u''}y-\frac{u'}{2y^{3/2}}\right)=\boxed{0}.
$$
Thus every twice-differentiable profile $u(\xi)$ gives a solution on $y>0$.

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