= Solution
The forcing contains $x/y$, so matching it with a single similarity variable requires $\beta=1$. For $u=y^\alpha f(\eta)$, $\eta=x/y$, substitution gives
$$
xu_{xx}-u_y=y^{\alpha-1}[\eta f''+\eta f'-\alpha f].
$$
Matching the source $y^{-1}\eta e^{-\eta}$ selects $\boxed{\alpha=0,\ \beta=1}$ and gives $f''+f'=e^{-\eta}$. Multiplying the first-order equation for $f'$ by $e^\eta$ yields $f'=(\eta+C)e^{-\eta}$, so $f=-(\eta+1+C)e^{-\eta}+D$. The limits at zero and infinity give $D=0$, $C=-1$. Hence
$$
\boxed{u_0(x,y)=-\frac xy e^{-x/y}.}
$$
For fixed positive $x$, the initial limit is also zero because the exponential dominates $x/y$. The function $-\eta e^{-\eta}$ decreases to $-1/e$ at $\eta=1$ and increases thereafter. Therefore
$$
\boxed{\min u_0=-1/e,\qquad\text{attained at every }x=y>0.}
$$
The axis limits are zero, but there is no joint limit at $(0,0)$: approaching along $x=y$ gives $-1/e$. This does not violate any of the printed fixed-variable limits. It is this bounded <similarity solution> whose minimum was calculated; the unbounded corner-source additions in (b) need not have the same minimum.
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