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

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

for the inner solution... I let

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

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

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

implying that

$\displaystyle \alpha =\frac{1}{3}$


Thanks in advance...