How do I impose initial conditions to solve a nonlinear ode in Mathematica?

Clear["Global`*"]

eqn = {u''[x] + x/2 u'[x] == 0, u[0] == u0};

sol = DSolve[eqn, u, x][[1]]

(* {u -> Function[{x}, u0 + Sqrt[π] C[1] Erf[x/2]]} *)

eqn /. sol

(* {True, True} *)

u[Infinity] /. sol

u0 + Sqrt[π] C[1]

For u to go to 0 at infinity, set the arbitrary constant to C[1] -> -u0/Sqrt[Pi]

sol2[x_, u0_] = (u /. sol /. C[1] -> -u0/Sqrt[Pi])[x] // FullSimplify

(* u0 Erfc[x/2] *)

sol2[Infinity, u0]

(* 0 *)

sol2[-Infinity, u0]

(* 2 u0 *)

Plot[Evaluate@Table[sol2[x, u0], {u0, 5, 1, -1}],
 {x, -4, 5}, PlotLegends -> Range[5, 1, -1]]