I have the following nonlinear ODE symmetric in the functions A(r) < - > B(r):
eq:= diff(A(r),r)^2/A(r)^2 + diff(A(r),r)*diff(B(r),r)/A(r)/B(r) + diff(B(r),r)^2/B(r)=k;
The task is to solve for A(r) in terms of B(r) and it goes like this: solve the quadratic equation for A'/A in terms of B'/B (or vise versa cause equation is symmetric) and then integrate.
dsolve(eq, A(r)); produces the required solution for A(r) in terms of integral over B and B'.
dsolve(eq, B(r)); returns nothing.
That must be a bug cause the equation is symmetric and solution for A in terms of B must have exactly the same functional form as solution for B in terms of A and is as easy to find. I use Maple 9.5.