Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later

Also posted in Wolfram Community.

I tried to solve the following nonlinear recurrence relation using RSolve.

y[n+1] = y[n] - a - b Sqrt[y[n]]

But the solution given by RSolve does not satisfy the relation.

Have I missed anything regarding RSolve or is this a bug in Mathematica?

Code is given below.

RSolve[y[n + 1] ==  y[n] - a - b Sqrt[y[n]], y, n]

FullSimplify[y[n + 1] == y[n] - a - b Sqrt[y[n]] /. %]

Sample Output

  • 2
    $\begingroup$ even making it more specific, we get a clearly wrong result, eg : RSolve[{y[n + 1] == y[n] + 1 + Sqrt[y[n]], y[0] == 1}, y, n] (I'm pretty sure there is not a closed form solution, but obviously we shouldn't get an incorrect solution) $\endgroup$
    – george2079
    Jul 29, 2016 at 14:28
  • 1
    $\begingroup$ When you crosspost at Community, please have the courtesy to mention that you posted at SE, and vice versa. $\endgroup$ Jul 29, 2016 at 18:08
  • $\begingroup$ @J.M., I apologise for my mistake. Thanks for pointing out. $\endgroup$ Jul 29, 2016 at 18:17
  • $\begingroup$ Considering Devendra's answer, can anybody with access to earlier versions add the customary header? $\endgroup$ Jul 29, 2016 at 20:19

1 Answer 1


This is a bug in RSolve.

The problem appears to be caused by a missing check while applying a transformation to solve this example, which should have returned unevaluated with the available methods.

Sorry for the confusion caused by the incorrect answer.


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