Incorrect result of DSolve

Question

Let us dsolve that Cauchy problem with 12.3 on Windows 10 Pro:

ClearAll[w, z]; sol = DSolve[{w'[z] == -1/2 - Sqrt[1/4 - 3*w[z]], w[1] == -1}, w[z], z]

{{w[z] -> 1/12 (-2 ProductLog[-((1 - Sqrt[13]) E^(2 + Sqrt[13] - 3 z))] - ProductLog[-((1 - Sqrt[13]) E^(2 + Sqrt[13] - 3 z))]^2)}, {w[ z] -> 1/12 (-2 ProductLog[(-1 + Sqrt[13]) E^( 2 + Sqrt[13] - 3 z)] - ProductLog[(-1 + Sqrt[13]) E^(2 + Sqrt[13] - 3 z)]^2)}}

and a warning "Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information". Both results are not correct as

Plot[w[z] /. sol[[1]], {z, 1, 2}]
Plot[w[z] /. sol[[2]], {z, 1, 2}]

show: w'[-1]==-1/2-Sqrt[13/4], making use of {w'[z] == -1/2 - Sqrt[1/4 - 3*w[z]], w[1] == -1}, is negative, whereas we see a positive slope at x==1 in the plots. What is the reason of that incorrect result and how to fix it?

Answer 1

Too long for a comment--not an answer.

Note that clearing w, doesn't clear w'[z].

DSolve does appear to be giving an invalid solution here. Perhaps, it is picking an incorrect branch.

sol = DSolve[{w'[z] == -1/2 - Sqrt[1/4 - 3*w[z]], w[1] == -1}, w[z], 
  z] 

Gives identical solutions: -(1/12) ProductLog[(-1 + Sqrt[13]) E^(2 + Sqrt[13] -3 z)] (2 + ProductLog[(-1 + Sqrt[13]) E^(2 + Sqrt[13] - 3 z)]

solutions = Simplify[w[z] /. sol]

Check ODE

Simplify[D[solutions[[1]], z] - (-1/2 - Sqrt[1/4 - 3*solutions[[1]]]),
  Assumptions -> z \[Element] Reals] (*not zero*)

I think this qualifies as a bug and should be reported.