I would like to solve differential equation derived from differentiating $u_i(q_1,q_2)=q_ip(q_1+q_2)-c_i(q_i)$ by $q_1$ and $q_2$, resp. and putting them equal to zero,and taking $q_2$ to be $r_2(q_1)$,the reaction function.
So we have a DE
$p(q_1+r_2(q_1))+p'(q_1+r_2(q_1))\cdot r_2(q_1)-c_2'(r_2(q_1))=0$
where $c_i(q_i)=c\cdot q_i$ for a fixed $c \in [0,1]$ and $p(q)=\max\{0,1-q\}$.
And we would like to compute $r_2:[0,\infty)\to [0,\infty)$.
What is the type of this ODE called, and how is it solved for $r_2$?
The solution should be $r_2(q1)=(1-q_1-c)/2$.