@matja You asked
...which differs from mathematica in the particular solution, there is on more term, namely: 2*r*exp(-r)*ln(2) which is already covered in the term r exp(-r)*_C2 of the general solution. What could be said to this?
If you put the new differential equation into the worksheet that I posted and run through the steps, then I think you'll see what's going on. The integration generates a term -ln(2*r), which becomes -ln(2) - ln(r) on simplification. One way to think of it is that the indefinite integral produces a constant of integration, but the int command does not supply an arbitrary constant . Since there are already two constants provided by dsolve, it is necessary that this extra constant matches up with a term which already has one of the dsolve-supplied constants. I guess that this can be used as a sort of accuracy check.
Maple's indefinite integration usually, but not always, produces 0 as the constant of integration; it depends on what is convenient. Here's a simple example, pretty close to the case at hand, where it doesn't.
Clearly 1/2*ln(2)^2 is a more convenient constant than 0 for the unexpanded form in this case.
Plaese let me know whether this explanation satisfies you.