# Simplifying expression using asymptotic values of a function

I have a large expression with bessel function in the result of DSolve. The equation is

DSolve[3/2 (-2 + x) x \[Alpha]^2 - y[x] + 2 Derivative[y][x] +
2 x (y^\[Prime]\[Prime])[x] == 0, {y[x]}, {x}]


Now, I am only bothered with the large x value of the solution. We know, bessel functions have simplified behavior in the asymptotic limit, like

$$I_n(x)=\frac{e^x}{\sqrt{2 \pi x}}$$.

Using this sort of simplified expressions, is it possible to reduce the result of DSolve using some predefined mathematica syntax? I am not able to do this using the "Series" command, as that gives a series with integer powers, and here, it may not be integers.

• Have you tried AsymptoticDSolveValue[]? Jul 7 at 4:44

Replacing $$x\rightarrow t^2/2$$ and using Series[...,{t,Infinity,0}] and using FullSimplify[...,Assumptions->t>0] I am getting this result
$$\frac{e^t \left(c_1-18 \alpha ^2\right)}{\sqrt{2 \pi t} }+\frac{e^{-t} \left(-18 i \alpha ^2+i c_1+2 \pi c_2\right)}{\sqrt{2 \pi t} }+\frac{3 \alpha ^2 t^4}{8}+\frac{3}{4} i \alpha ^2 e^{-2 t} t^3$$