Maple 16's solution (switching to lowercase because I has a special meaning in Maple):
e1:= i(t) = c*diff(v(t),t); e2:= v(t)*i(t)=p+r[c]*i(t)^2; simplify(dsolve({e1,e2}));
$\displaystyle [ \left\{ v \left( t \right) =r_{{c}}p \left( {\it LambertW} \left( -4\,r_{{c}}p{{\rm e}^{{\frac {-2\,t+r_{{c}}c+2\,{\it \_C1}}{r_{{c}}c}}}} \right) -1\\ \mbox{} \right) \left( \sqrt{-r_{{c}}p \left( {\it LambertW} \left( -4\,r_{{c}}p{{\rm e}^{{\frac {-2\,t+r_{{c}}c+2\,{\it \_C1}}{r_{{c}}c}}}}\\ \mbox{} \right) \right) ^{-1}} \right) ^{-1} \left( {\it LambertW} \left( -4\,r_{{c}}p{{\rm e}^{{\frac {-2\,t+r_{{c}}c+2\,{\it \_C1}}{r_{{c}}c}}}} \right) \right) ^{-1},v \left( t \right) =-r_{{c}}p \left( -1+{\it LambertW} \left( -1/4\,{{\rm e}^{{\frac {-2\,t+r_{{c}}c+2\,{\it \_C1}}{r_{{c}}c}}}}{r_{{c}}}^{-1}{p}^{-1} \right) \right) \sqrt{-{r_{{c}}}^{-1}{p}^{-1} \left( {\it LambertW} \left( -1/4\,{{\rm e}^{{\frac {-2\,t+r_{{c}}c+2\,{\it \_C1}}{r_{{c}}c}}}}{r_{{c}}}^{-1}{p}^{-1} \right) \right) ^{-1}}\\ \mbox{} \right\} , \left\{ i \left( t \right) =c{\frac {d}{dt}}v \left( t \right) \right\} ] $
It may actually be better to use the implicit form:
dsolve({e1,e2},implicit);
$\displaystyle [ \left\{ t-1/4\,{\frac {c \left( v \left( t \right) \right) ^{2}}{p}}-1/4\,{\frac {cv \left( t \right) \sqrt{ \left( v \left( t \right) \right) ^{2}-4\,r_{{c}}p}}{p}}+r_{{c}}c\ln \left( v \left( t \right) + \sqrt{ \left( v \left( t \right) \right) ^{2}-4\,r_{{c}}p} \right) \\ \mbox{}-{\it \_C1}=0,t-1/4\,{\frac {c \left( v \left( t \right) \right) ^{2}}{p}}+1/4\,{\frac {cv \left( t \right) \sqrt{ \left( v \left( t \right) \right) ^{2}-4\,r_{{c}}p}}{p}}-r_{{c}}c\ln \left( v \left( t \right) + \sqrt{ \left( v \left( t \right) \right) ^{2}-4\,r_{{c}}p} \right) \\ \mbox{}-{\it \_C1}=0 \right\} , \left\{ i \left( t \right) =c{\frac {d}{dt}}v \left( t \right) \right\} ]$