I have a question (probably on the dumber side...sorry) I hope someone can help...

The problem :
Find a uniform approximation for:
\epsilon u'' +u'+xu=0

for the inner solution... I let

\bar{x} =\frac{x}{\epsilon^{\alpha}}   \hspace{3mm}u(x)=U

My question is this... I want to show 1~2 balance by ruling out others... so in the case 1~3 balance... are the coef going to be


for 1
\epsilon^{1-2\alpha}\hspace{1mm}U'

for 3
\epsilon^{\alpha} \bar{x}\hspace{1mm}U\\

implying that

\alpha =\frac{1}{3}


Thanks in advance...