Got it, I tried to explore my manual calculation, here it is

diffeq:={diff(a(r),r)=a(r)+b(r)+c(r),diff(b(r),r)=a(r)-b(r)+c(r),diff(c(r),r)=a(r)-b(r)-c(r)};

dsolve(diffeq,{a(r),b(r),c(r)});

use DEplot to perform the plot..finished!

DEplot(......)