# DSolve Returns Multiple Solutions to a Well-Posed Initial Value problem

The initial value problem: $dy/dx=x\sqrt{y},\;y(0)=1$ has the unique solution$$y=\frac 1{16}(x^4+8x^2+16)$$Mathematica's DSolve returns two solutions; the above and$$y=\frac 1{16}(x^4-8x^2+16)$$This second "solution" is erroneous.

• Next time, please include explicit Mathematica code: DSolve[{y'[x] == x Sqrt[y[x]], y[0] == 1}, y, x]. I believe this is a known long-standing bug in DSolve[] (but I'll let someone else add the bugs tag). Here is a more striking example. Aug 25, 2017 at 3:16
• Nearly the same: mathematica.stackexchange.com/questions/86665/… Aug 25, 2017 at 11:31

eqns = {y'[x] == x Sqrt[y[x]], y[0] == 1};

sol = DSolve[eqns, y, x]

(*  {{y -> Function[{x}, 1/16 (16 - 8 x^2 + x^4)]}, {y ->
Function[{x}, 1/16 (16 + 8 x^2 + x^4)]}}  *)


The second solution is valid for all real x

eqns /. sol // FullSimplify[#, Element[x, Reals]] &

(*  {{x^3 == x (4 + Abs[-4 + x^2]), True}, {True, True}}  *)

Reduce[x^3 == x (4 + Abs[-4 + x^2]), x, Reals]

(*  x <= -2 || x == 0 || x >= 2  *)


The first solutions is valid on the domain x <= -2 || x == 0 || x >= 2

eqns /. sol // FullSimplify[#, x <= -2 || x == 0 || x >= 2] &

(*  {{True, True}, {True, True}}  *)