Doubts about change of variable in ODE

[arroy_0205]

I was trying to check some calculations from a paper myself and I got
stuck at a differential equation. Can anybody help me with suggestions?
I am giving a simpler version of the differential eqn by setting all numerical
constants=1. Consider:

e^{-2y}\left(z''(y)-2z'(y)+\frac{3}{4}z(y)\right)+z(y)=0

We change variable to

u=e^y

then the ODE is

u^2z''(u)-uz'(u)+z(u)(u^2+\frac{3}{4})=0

The author writes the solution is:

z(y)=\frac{1}{u^{3/2}}z(u)=u\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)

where J and N are Bessel functions. My problem is, I am getting the relation

z(u)=u\left(c_1 J_{\frac{1}{2}}(u)+c_2 N_{\frac{1}{2}}(u) \right)

But note that according to the author,

z(y)={u^{5/2}}\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)

Or in other words though I can get the desired solution in terms of changed variable,
I am not able to get the relation:

z(y)=\frac{1}{u^{3/2}}z(u)

at all. How to get this relation? Also, note that the author uses one constant
in the solution but I think there should be two independent constants. What am I missing in the calculations?