I am currently doing a Masters-level class in Perturbation Methods and I am stuck on a question. I have done similar questions but the middle term is confusing me. It could be that I'm rescaling wrong maybe?
The problem is Exercise 5.5 from Perturbation Methods by E. J. Hinch. Here it is:
The function $y(x;\epsilon)$ satisfies
$$
\epsilon y'' + x^{1/2} y' +y=0 \quad \quad \text{in } 0 \leq x \leq 1
$$
and is subject to boundary conditions $y=0$ at $x=0$ and $y=1$ at $x=1$. First find the rescaling for the boundary layer near $x=0$, and obtain the leading order inner approximation. Then find the leading order outer approximation and match the two approximations.
Best Answer
Changing coordinates to $x=δX$ and $Y(X)=y(δX)$ gives the scaled equation $$ ϵY''(X)+(δX)^{1/2}δY'(X)+δ^2Y(X)=0. $$ The scale factors that you need to balance are thus $ϵ,δ^{3/2},δ^2$, which gives the scaling for the inner solution at $ϵ=δ^{3/2}$ resp. $δ=ϵ^{2/3}$, and the equation to solve as $$ Y''(X)+X^{1/2}Y'(X)=0 $$