Question: How Can I solve a nonlinear edo and compare with some data ?

I am primer with maple.

I have the following  nonlinear equation.

EQ:=diff(N(t),t) -kappa/(q-1)*( 1- (N(t)/K )^(q) - (1-N(t)/K)^(q))=0;

As, we see this equation is high non linear , We can solve for some specific : kappa , K and q. (These are parameters)

The problem is that.  I want to compare my solution with some  data. In order to validate this model. For example, I give some expecific initials conditions and some values for kappa, K and q. I did use the following parameters: q = 0.94, K = 2.2 , kappa = 0.06314 and the initial condition is N(t= 13.2621)= 0.0012 ;
 

And  this is the experimetal data:

X:=VectorApplyFunction( [0.0012  0.0017 0.0031 0.0063  0.0174 0.0317 0.0598 0.0821 0.1501 0.2061 0.2742 0.3534 0.4852 0.5337 0.6059
  0.7565 1.0723  1.1068  1.2173]  ,datatype=float);

Y:=VectorApplyFunction( [13.2621   14.0740 15.4272  16.7803   18.1335 19.2161 20.5692 21.6518 23.0049 24.0875 25.1700 26.5232  27.3351 28.6883 29.2295                          
30.5827   32.4772  34.1010 34.6422 ]  ,datatype=float); 
 

I want to know the best values for kappa, K, and q.
Please Let me know, How Can I do for solve this problem.

Thanks in Avance.