How to solve this second order differential equation?

Question

The equation is given by:

$$ \frac{dv}{du}\left(2 u\frac{1-uv}{1+uv} + u\right) = \frac{v}{2} \frac{d^2v}{du^2} + v\frac{1-uv}{1+uv} $$

I tried using DSolve for this differential equation, but it's not giving me any answer.

DSolve[
  v'[u] (2 u (1 - u v[u])/(1 + u v[u]) + u) ==  
    v[u] (1 - u v[u])/(1 + u v[u]) + v''[u] (v[u]/2), 
  v[u], u]

How do I solve this?

Edit

In case you want to solve the equation numerically, the boundary conditions are

$$v\left(\frac{1+\sqrt{4.9}}{2}\right) = \frac{-2}{1 + \sqrt{4.9}}$$ $$v\left(\frac{1-\sqrt{4.9}}{2}\right) = \frac{-2}{1 - \sqrt{4.9}}$$

Answer 1

In case you want to solve the equation numerically, the boundary conditions are

Your boundary conditions are not consistent with the ODE. Did you check?

Looking at this term in your ODE

 s = (1 - u*v[u])/(1 + u*v[u]);

Then you say one boundary condition is

 z = Sqrt[49/10];
 (v[(1 + z)/2] == -2/(1 + z)) // N

Lets check what happens to s at the above boundary condition

 s /. {u -> (1 + z)/2, v[u] -> -2/(1 + z)}

The same for the other boundary condition

 s /. {u -> (1 - z)/2, v[u] -> -2/(1 - z)}

So you can not solve it even numerically unless you fix the boundary conditions.