eqn = x'[t] == Sqrt[x[t]];
soln = DSolve[{eqn, x[0] == x0}, x[t], t] // FullSimplify
{{x[t] -> 1/4 (t - 2 Sqrt[x0])^2}, {x[t] -> 1/4 (t + 2 Sqrt[x0])^2}}
both solutions satisfy the initial condition
soln /. t -> 0
{{x[0] -> x0}, {x[0] -> x0}}
rules = Flatten[NestList[D[#, t] &, #, 1]] & /@ soln
{{x[t] -> 1/4 (t - 2 Sqrt[x0])^2, Derivative[1][x][t] -> 1/2 (t - 2
Sqrt[x0])}, {x[t] -> 1/4 (t + 2 Sqrt[x0])^2, Derivative[1][x][t]
-> 1/2 (t + 2 Sqrt[x0])}}
The second solution is valid for (x0 >= 0) and (t >= -2 Sqrt[x0])
Simplify[eqn /. rules, {x0 >= 0, t >= -2 Sqrt[x0]}]
{t >= 2 Sqrt[x0], True}
Both solutions are valid for (x0 >= 0) and (t >= 2 Sqrt[x0])
Simplify[eqn /. rules, {x0 >= 0, t >= 2 Sqrt[x0]}]
{True, True}