I missed that, thank u very much !

Here is the worksheet:

> a := 0.152922e-4;
> b := 1.62281;
> PCO2 := 0.22e8;
> sigmaCO2 := 0.30e-1;
> rp := 0.10e-4;
> f := x-> sigmaCO2*sqrt((36-12*Pi/sqrt(3))/x)/rp;
> n := 1;
> T := 313;
> K := x->a*(PCO2-f(x))+b;
> sigmas := 0.38e-1;
> ups := 0.27e-4;
> R := 8.314;
> equ := 0 = -32*sigmas^2*ups*rp/(R*T)^2*(2/ln(S(x))^3).(diff(ln(S(x)), x))+(1/2)*x*(sqrt(K(x)/S(x))*(diff(S(x), x))+sqrt(S(x)/K(x))*(D(K))(x))+sqrt(S(x)*K(x));
> ics := S(.5) = .1;
> solution := dsolve({equ, ics}, numeric, range = 0.1e-2 .. .5);

I think the error may come from the definitions of the functions K and f.