TITLE: Closed form of the solution of a nonlinear differential equation
QUESTION [2 upvotes]: I should solve the following problem: given a function $u(x)$, the sum of the function and its reciprocal must be equal to the integral of the function raised to $k$. Taking the derivative of the two members, left and right, I get the following differential equation:
$$\frac{d}{dx}\left(u(x)+\frac{1}{u(x)}\right)=u(x)^k$$
that can be transformed in:
$$u(x)'\left[ 1-\frac{1}{u(x)^2}\right]-u(x)^k=0$$
Its solution can be expressed in closed form for $k=1$ using the $LambertW$ function.
Also for $k=2$ and $k=3$ there are solutions in a closed form. Is it possible to find a closed form solution for every $k\in \mathbb{N}$?
Thanks.

REPLY [0 votes]: Write the equation as 
$$
\frac{u'}{u^k}-\frac{u'}{u^{k+2}}=1.
$$
Integrate to get
$$
\frac{1}{(1-k)u^{k-1}}+\frac{1}{(k+1)u^{k+1}}=x-x_0,
$$
where $x_0$ is a free constant. This relation can be rewritten as
$$
u^{k+1}(x_0-x)(1-k^2)+u^2(k+1)+1-k=0.
$$
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