It is known that a zero-order Bessel equation can be written in the form:
$$\frac{d^2U}{dx^2}+\frac{1}{x}\frac{dU}{dx}+\beta^2U=0$$
having solution:
$$U(x) = C_1 J_0(\beta x)+C_2 Y_0(\beta x)$$
where $J_0$ and $Y_0$ are Bessel functions of first and second kind, respectively, and $C_1, C_2$ being constants.
Now, for the folowing equation, with a parameter $\alpha$:
$$\frac{d^2U}{dx^2}+\frac{\alpha}{x}\frac{dU}{dx}+\beta^2U=0$$
How the solution $U(x)$ can be written?