This ODE has one general solutions or two solutions? The ODE
$$2xy''-y'+\frac{1}{y'}=0$$
is a second order ODE , so it must have one general solution containing 2 constants  .
I solved it by reduction of order as following but I have 2 general solutions each contains 2 constants , since I have positive and negative square root . I would like to know , do they represent one general solution since the sign can be included into the constant ? or do they represent as I said before 2 general solutions ?
Here is my solution :
let v=y' , v'=y''
$$2xv'=v-\frac{1}{v}$$
$$v^2-1=cx$$
$$v=\pm\sqrt{cx+1}$$
$$\frac{dy}{dx}=\pm\sqrt{cx+1}$$
$$y=\pm\frac{2}{3c}(cx+1)^{3/2}+k$$
 A: $\frac{dy}{dx}=\pm\sqrt{cx+1}$ is a valid equation, but it is not a valid expression for a function, and therefore not integrable. Read $\pm$ as plus or  minus, not plus and minus. The derivative must be equal to either the positive or negative portion. You would solve this by using your initial conditions of the ODE.
A: You have not 2 general solutions each contains 2 constants. You have two sub-sets of solutions. Each one in not a general solution. The general solution is the gather of the two.
Another expression of the general solution is on the form of an implicit equation :
$$4(cx+1)^3-9c^2(y-k)^2=0 \qquad\text{There is no }\pm\text{ in it.}$$ $$ $$
The situation is comparable to this :
A circle which equation is :
$$x^2+y^2=r^2$$
is equivalently described by :
$$y(x)=\pm\sqrt{r^2-x^2}$$
which means the gather of to semicircles $\quad y(x)=\sqrt{r^2-x^2}\quad $ and $\quad y(x)=-\sqrt{r^2-x^2}$
$$ $$
In fact this is not yet all the solutions of the PDE. 
When you wrote : 
" $2xy″−y′+1y′=0$ is a second order ODE , so it must have one general solution containing 2 constants" 
you are thinking of a linear ODE. But the ODE is not linear. So, the above statment is false. 
The solutions of the ODE are not only 
$$\quad y=\pm\frac{2}{3c}(cx+1)^{3/2}+k\quad $$ 
but also 
$$y(x)=\pm x+k$$
This can be seen directly from the ODE in case of $y''=0$ or alternatively in looking for the envelope of the family of curves $4(cx+1)^3-9c^2(y-k)^2=0$ which differentiation wrt $c$ leads to $c=2/x$ and then $x^2-(y-k)^2=0$.
