Question on proper technique for solving this equation

[redtree]

This is NOT a homework question. I am having trouble solving the following differential equation to solve for A[r]:

Any suggestions?

 

[fresh_42]

As far as I can see it is a Bernoulli differential equation of the form $A'(r) = p(r) A^2(r) + q(r) A(r)$ with rational functions $p(r)$ and $q(r)$. The substitution $B(r) = A^{-1}(r)$ delivers then the LDE $B' = -p-q B$.

 

[redtree]

Awesome! That's it. Thanks so much.