Determining an exponent

Question

I have the expression $$C_1x^a+C_2 yx^{-b-1/2}\qquad(\star)$$ where $C_1\ne C_2>0$

I want to choose $x\sim y^\Box$, where $\Box$ is an unknown exponent. My question is, how can I choose $\Box$ so that $$(\star)\sim C_3y^\Diamond$$ where $C_3>0$ and $\Diamond$ is some other exponent (in terms of $a$ and $b$), yet to be determined?

Note that for any $x,y$, we have that $x\sim y$ means that they are of the same order

I have a suspicion that $\Diamond=(a/b+1/2b)/(a/b+1/2b+1)$, but I'm not sure

Answer 1

For the first two terms to be homogeneous, you need $y\propto x^{a+b+1/2}$.

Then

$$x\propto y^{1/(a+b+1/2)}$$ and $$x^a\propto y^{a/(a+b+1/2)}.$$