ClearAll["Global`*"]
\[Nu] = 5*10^-5;
k = 2*\[Pi]*\[Nu];
sol = NDSolve[{\[Nu]*u''[y] + k*(y)^2*Abs[u'[y]]*u''[y] == 0, 
   u[0] == 0, u'[0.999553801] == 0}, u, {y, 0, 0.999553801}] 

I cant understand my error it says NDSolve::ndode: The equations {0[0.999554]==0} are not differential equations or initial conditions in the dependent variables {u}.

Jose,

2 problems with your update. You have a typo in 2 places: y^'' should just be y''

Also, you should not specify initial conditions on x''' because it is determined from an equation. Same for y''. Both are not valid initial conditions. (see @Gianluca Gorni post above)

This works: (as would any combination of initial conditions that does not involve x''' nor y'')

NDSolve[{x'''[t] == x[t] Derivative[1][y][t], 
  y''[t] == -(1/(x[t]^2 + y[t]^2)), x[0] == 1, x''[0] == 0, 
  y[0] == 1/8, Derivative[1][y][0] == 0, y'[0] == 1/2}, {x, y}, {t, 0,
   1}]

Regards

Thank you.. I made up the DE problem . I am looking to test how NDsolve does works. I have made another correction but still don't know why I am no getting anywhere.