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1. Looking for a function with specific properties

Hi everyone, I'm trying to find a function of single variable f(x) with the following properties: -It is symmetric around zero -It is differentiable everywhere -f'(x)≥0 for all x>0 -f'(x)=0 when x=0 -f'(x)≤0 for all x<0 (I think these last two actually follow from the first three?)...
2. Does the series k a^k have a name?

Hi, a naive question here, but I was wondering if the series \sum^{∞}_{k=0} k a^{k} has a particular name? As in 'geometric series' for \sum^{∞}_{k=0} a^{k} ? And what about the more general \sum^{∞}_{k=0} k^{n} a^{k} ? As a related question, you seem to be able to get the formula for the...
3. Integrating directly from a pair of diff.eqs. without solving them

This is actually related to a post I made earlier in the differential equations forums, but I've since realized that solving the equations themselves is not necessarily the best way to get where I want to go. Perhaps it's better suited to this forum, since it is an integration problem that I...
4. Integrating 1/(x+ln(x))

I've been trying to figure out how to integrate 1/(x+ln(x)) but am not getting anywhere. Mathematica can't do it, and I haven't found it in lists of integrals. Does anyone know if this integral exists in closed form? Same goes for (x+ln(x))/(1+x+ln(x)) Thanks! zeroseven
5. Solving a pair of nonlinear coupled DEs

Hi everyone, new member zeroseven here. First, I want to say that it's great to have a forum like this! Looking forward to participating in the discussion. Anyway, I need to solve a pair of differential equations for an initial value problem, but am not sure if an analytical solution exists. I...