I'm hoping to get some help answering the following question. I've seen it asked before, but I sadly didn't understand any of the responses.
Find the first two terms in an asymptotic expansion in powers of the small parameter ε of the solution of:
$$ xy'(x) + y(x) = ε(y(x))^\frac 12, x > 1, y(1) = 1 $$
Here's what I've done so far:
Letting ε = 0, we obtain $$ xy'(x) + y(x) = 0 $$
hence $$ y(x) = \frac cx $$
c constant.
Using y(1) = 1, we get $$ y(x) = \frac 1x $$
Now I assume a better approximation of the form $$ y(x) = \frac 1x + εy_0(x) + ε^2y_1(x) + O(ε^3) $$
I can differentiate this to get $$ y'(x) = -(\frac 1x)^2 + εy_0'(x) + ε^2y_1'(x) + O(ε^3) $$
In other examples, I have then gone on to plug y(x) and y'(x) into the original ODE, then equate the coefficients of each power of ε to find y0 and y1, but I'm not sure how to deal with the square root on the RHS of the ODE.
i.e. how do I write the RHS as a polynomial in ε?
Any help would be appreciated, thank you!