Generalizing the solution to an ODE Is there a way to solve the following ODE for general integral values of $m$
\begin{align}
\frac{\partial A(x)}{ \partial x} = -A(x)^m + \frac1x  \label{rec}\tag{1}
\end{align}
I have some ways to approach this problem for a special case of $m=2$. For this case, if we substitute $A(x) = \frac{u^\prime(x)}{u(x)}$, we would get a differential equation of the form,
$$u^{\prime\prime}(x) = \frac{u(x)}x$$
And, it is possible to write a solution for this equation in terms of Bessel functions. But I don't know how to generalize this for higher $m$.
Any help would be appreciated!

This is a Chini differential equation, but the Chini invariant depends on $x$.  I don't think
a closed-form solution is known in general.  Maple does find the general solution for $n=2$:
$$ A \left( x \right) =-{\frac {-c {{ I}_{0}\left(2\,\sqrt {x}\right)}+{{ K}_{0}\left(2\,\sqrt {x}\right)}}{\sqrt {x} \left( {
\it c}\,{{ I}_{1}\left(2\,\sqrt {x}\right)}+{{ K}_{1}\left(2
\,\sqrt {x}\right)} \right) }}
$$
but not for $m=3$.
