how to solve this non linear ode

Question

$$ y y'(x) +y(x)^2(\sqrt {x^3}+{7\over4}\sqrt {x^5}+{1\over2}\sqrt {x^7})-{1\over2x}=0 $$

How to solve this equation?? I searched text book , and I only found bessel, legandre. But they are not same shape with my equation. What should I do for solving this equation?

Answer 1

If you divide the equation by $y$, then you see that the equation is a bernoulli differential equation. You can see a general idea to solve such a differential equation in the following page. http://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx

Answer 2

A term of the solution of the EDO involves an integral which cannot be expressed on closed form with the standard functions (Analytical solution in attachment).