I need to show that the leading order inner solution is given by the below. Thus far, I have rescaled and showed the boundary layer is of order $\epsilon^{\frac{3}{4}}$. Hence at leading order I then try to solve the second order ode $Y_0''(X) + X^{2}Y_0'(X)=0$ but don't get the result. Please help
Here is what I did, our layer thicknesses don't agree:
We know the boundary layer is at $x = 0,$ so let $\xi = x \epsilon^{-a}.$ Then the equation is, $$\epsilon^{1-2a} y''(\xi) + \xi^2 \epsilon^a - \xi^3 \epsilon^{3a} y = 0.$$
If $a = 1/3,$ then the first two terms balance and the third term is of higher order, so the thickness is ${\cal O}(\epsilon^{1/3}).$ The equation has now become, $$ y'' + \xi^2 y' = 0.$$
Doing the typical $y(\xi) \sim Y_0(\xi) + \cdots,$ the equation you stated is now the problem to solve. Clearly one may integrate this straight up one time to get $$Y'_0 = c_2 e^{-\xi^3 / 3}$$ This is just your basic use of an integrating factor, $e^{\xi^3/3}.$ So then integrate it one more time to get $Y_0(\xi) = c_1 + c_2\int_0^\xi e^{-t^3/3} \; dt.$